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(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

Require Import Rbase.
Require Import Rfunctions.
Require Import Ranalysis_reg.
Require Import Classical_Prop.
Local Open Scope R_scope.

Set Implicit Arguments.

(*****************************************************)

Each bounded subset of N has a maximal element

(*****************************************************)

Definition Nbound (I:nat -> Prop) : Prop :=
  exists n : nat, (forall i:nat, I i -> (i <= n)%nat).

Lemma IZN_var : forall z:Z, (0 <= z)%Z -> {n : nat | z = Z.of_nat n}.forall z : Z, (0 <= z)%Z -> {n : nat | z = Z.of_nat n}
Proof.forall z : Z, (0 <= z)%Z -> {n : nat | z = Z.of_nat n}
  intros; apply Z_of_nat_complete_inf; assumption.
Qed.

Lemma Nzorn :
  forall I:nat -> Prop,
    (exists n : nat, I n) ->
    Nbound I -> { n:nat | I n /\ (forall i:nat, I i -> (i <= n)%nat) }.forall I : nat -> Prop,
(exists n : nat, I n) ->
Nbound I ->
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
Proof.forall I : nat -> Prop,
(exists n : nat, I n) ->
Nbound I ->
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  intros I H H0; set (E := fun x:R =>  exists i : nat, I i /\ INR i = x);
    assert (H1 : bound E).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x : R => exists i : nat, I i /\ INR i = x:R -> Prop
bound E
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x : R => exists i : nat, I i /\ INR i = x:R -> Prop
H1:bound E
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  unfold Nbound in H0; elim H0; intros N H1; unfold bound;
    exists (INR N); unfold is_upper_bound; intros;
      unfold E in H2; elim H2; intros; elim H3; intros;
        rewrite <- H5; apply le_INR; apply H1; assumption.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x : R => exists i : nat, I i /\ INR i = x:R -> Prop
H1:bound E
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  assert (H2 :  exists x : R, E x).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x : R => exists i : nat, I i /\ INR i = x:R -> Prop
H1:bound E
exists x : R, E x
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x : R => exists i : nat, I i /\ INR i = x:R -> Prop
H1:bound E
H2:exists x : R, E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  elim H; intros; exists (INR x); unfold E; exists x; split;
    [ assumption | reflexivity ].I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x : R => exists i : nat, I i /\ INR i = x:R -> Prop
H1:bound E
H2:exists x : R, E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  destruct (completeness E H1 H2) as (x,(H4,H5)); unfold is_upper_bound in H4, H5;
      assert (H6 : 0 <= x).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
0 <= x
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  destruct H2 as (x0,H6).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
x0:R
H6:E x0
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
0 <= x
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)} remember H6 as H7.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
x0:R
H6:E x0
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H7:E x0
HeqH7:H7 = H6
0 <= x
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)} destruct H7 as (x1,(H8,H9)).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
x0:R
H6:E x0
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
x1:nat
H8:I x1
H9:INR x1 = x0
HeqH7:ex_intro (fun i : nat => I i /\ INR i = x0) x1
  (conj H8 H9) = H6
0 <= x
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
    apply Rle_trans with x0;
      [ rewrite <- H9; change (INR 0 <= INR x1); apply le_INR;
        apply le_O_n
        | apply H4; assumption ].I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  assert (H7 := archimed x); elim H7; clear H7; intros;
    assert (H9 : x <= IZR (up x) - 1).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
x <= IZR (up x) - 1
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  apply H5; intros x0 H9.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
x0:R
H9:E x0
x0 <= IZR (up x) - 1
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)} assert (H10 := H4 _ H9); unfold E in H9; elim H9; intros x1 (H12,<-).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
x1:nat
H10:INR x1 <= x
H9:exists i : nat, I i /\ INR i = INR x1
H12:I x1
INR x1 <= IZR (up x) - 1
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
    apply Rplus_le_reg_l with 1;
      replace (1 + (IZR (up x) - 1)) with (IZR (up x));
      [ idtac | ring ]; replace (1 + INR x1) with (INR (S x1));
      [ idtac | rewrite S_INR; ring ].I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
x1:nat
H10:INR x1 <= x
H9:exists i : nat, I i /\ INR i = INR x1
H12:I x1
INR (S x1) <= IZR (up x)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  assert (H14 : (0 <= up x)%Z).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
x1:nat
H10:INR x1 <= x
H9:exists i : nat, I i /\ INR i = INR x1
H12:I x1
(0 <= up x)%Z
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
x1:nat
H10:INR x1 <= x
H9:exists i : nat, I i /\ INR i = INR x1
H12:I x1
H14:(0 <= up x)%Z
INR (S x1) <= IZR (up x)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  apply le_IZR; apply Rle_trans with x; [ apply H6 | left; assumption ].I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
x1:nat
H10:INR x1 <= x
H9:exists i : nat, I i /\ INR i = INR x1
H12:I x1
H14:(0 <= up x)%Z
INR (S x1) <= IZR (up x)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  destruct (IZN _ H14) as (x2,H15).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
x1:nat
H10:INR x1 <= x
H9:exists i : nat, I i /\ INR i = INR x1
H12:I x1
H14:(0 <= up x)%Z
x2:nat
H15:up x = Z.of_nat x2
INR (S x1) <= IZR (up x)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
    rewrite H15, <- INR_IZR_INZ; apply le_INR; apply lt_le_S.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
x1:nat
H10:INR x1 <= x
H9:exists i : nat, I i /\ INR i = INR x1
H12:I x1
H14:(0 <= up x)%Z
x2:nat
H15:up x = Z.of_nat x2
(x1 < x2)%nat
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
      apply INR_lt; apply Rle_lt_trans with x;
        [ assumption | rewrite INR_IZR_INZ; rewrite <- H15; assumption ].I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  assert (H10 : x = IZR (up x) - 1).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
x = IZR (up x) - 1
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  apply Rle_antisym;
    [ assumption
      | apply Rplus_le_reg_l with (- x + 1);
        replace (- x + 1 + (IZR (up x) - 1)) with (IZR (up x) - x);
        [ idtac | ring ]; replace (- x + 1 + x) with 1;
        [ assumption | ring ] ].I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  assert (H11 : (0 <= up x)%Z).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
(0 <= up x)%Z
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  apply le_IZR; apply Rle_trans with x; [ apply H6 | left; assumption ].I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  assert (H12 := IZN_var H11); elim H12; clear H12; intros x0 H8; assert (H13 : E x).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
E x
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  elim (classic (E x)); intro; try assumption.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
E x
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  cut (forall y:R, E y -> y <= x - 1).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
(forall y : R, E y -> y <= x - 1) -> E x
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
forall y : R, E y -> y <= x - 1
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  intro H13; assert (H14 := H5 _ H13); cut (x - 1 < x).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
H13:forall y : R, E y -> y <= x - 1
H14:x <= x - 1
x - 1 < x -> E x
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
H13:forall y : R, E y -> y <= x - 1
H14:x <= x - 1
x - 1 < x
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
forall y : R, E y -> y <= x - 1
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  intro H15; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H14 H15)).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
H13:forall y : R, E y -> y <= x - 1
H14:x <= x - 1
x - 1 < x
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
forall y : R, E y -> y <= x - 1
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  apply Rminus_lt; replace (x - 1 - x) with (-1); [ idtac | ring ];
    rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; apply Rlt_0_1.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
forall y : R, E y -> y <= x - 1
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  intros y H13; assert (H14 := H4 _ H13); elim H14; intro H15; unfold E in H13; elim H13;
    intros x1 H16; elim H16; intros H17 H18; apply Rplus_le_reg_l with 1.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
1 + y <= 1 + (x - 1)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y = x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
1 + y <= 1 + (x - 1)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  replace (1 + (x - 1)) with x; [ idtac | ring ]; rewrite <- H18;
    replace (1 + INR x1) with (INR (S x1)); [ idtac | rewrite S_INR; ring ].I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
INR (S x1) <= x
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y = x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
1 + y <= 1 + (x - 1)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  cut (x = INR (pred x0)).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
x = INR (Init.Nat.pred x0) -> INR (S x1) <= x
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
x = INR (Init.Nat.pred x0)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y = x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
1 + y <= 1 + (x - 1)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  intro H19; rewrite H19; apply le_INR; apply lt_le_S; apply INR_lt; rewrite H18;
    rewrite <- H19; assumption.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
x = INR (Init.Nat.pred x0)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y = x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
1 + y <= 1 + (x - 1)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  rewrite H10; rewrite H8; rewrite <- INR_IZR_INZ;
    rewrite <- (minus_INR _ 1).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
INR (x0 - 1) = INR (Init.Nat.pred x0)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
(1 <= x0)%nat
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y = x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
1 + y <= 1 + (x - 1)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  apply f_equal;
    case x0; [ reflexivity | intro; apply sym_eq, minus_n_O ].I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
(1 <= x0)%nat
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y = x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
1 + y <= 1 + (x - 1)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  induction x0 as [|x0 Hrecx0].I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
H8:up x = Z.of_nat 0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
(1 <= 0)%nat
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat (S x0)
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
Hrecx0:up x = Z.of_nat x0 -> (1 <= x0)%nat
(1 <= S x0)%nat
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y = x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
1 + y <= 1 + (x - 1)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
    rewrite H8 in H3.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:IZR (Z.of_nat 0) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
H8:up x = Z.of_nat 0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
(1 <= 0)%nat
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat (S x0)
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
Hrecx0:up x = Z.of_nat x0 -> (1 <= x0)%nat
(1 <= S x0)%nat
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y = x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
1 + y <= 1 + (x - 1)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)} rewrite <- INR_IZR_INZ in H3; simpl in H3.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x0 : R => exists i : nat, I i /\ INR i = x0:R -> Prop
H1:bound E
H2:exists x0 : R, E x0
x:R
H4:forall x0 : R, E x0 -> x0 <= x
H5:forall b : R,
(forall x0 : R, E x0 -> x0 <= b) -> x <= b
H6:0 <= x
H3:0 > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
H8:up x = Z.of_nat 0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
(1 <= 0)%nat
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat (S x0)
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
Hrecx0:up x = Z.of_nat x0 -> (1 <= x0)%nat
(1 <= S x0)%nat
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y = x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
1 + y <= 1 + (x - 1)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
      elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H6 H3)).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat (S x0)
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y < x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
Hrecx0:up x = Z.of_nat x0 -> (1 <= x0)%nat
(1 <= S x0)%nat
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y = x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
1 + y <= 1 + (x - 1)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
    apply le_n_S; apply le_O_n.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H12:~ E x
y:R
H13:exists i : nat, I i /\ INR i = y
H14:y <= x
H15:y = x
x1:nat
H16:I x1 /\ INR x1 = y
H17:I x1
H18:INR x1 = y
1 + y <= 1 + (x - 1)
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
    rewrite H15 in H13; elim H12; assumption.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x1 : R => exists i : nat, I i /\ INR i = x1:R -> Prop
H1:bound E
H2:exists x1 : R, E x1
x:R
H4:forall x1 : R, E x1 -> x1 <= x
H5:forall b : R,
(forall x1 : R, E x1 -> x1 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:E x
{n : nat |
I n /\ (forall i : nat, I i -> (i <= n)%nat)}
  split with (pred x0); unfold E in H13; elim H13; intros; elim H12; intros;
    rewrite H10 in H15; rewrite H8 in H15; rewrite <- INR_IZR_INZ in H15;
      assert (H16 : INR x0 = INR x1 + 1).I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:exists i : nat, I i /\ INR i = x
x1:nat
H12:I x1 /\ INR x1 = x
H14:I x1
H15:INR x1 = INR x0 - 1
INR x0 = INR x1 + 1
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:exists i : nat, I i /\ INR i = x
x1:nat
H12:I x1 /\ INR x1 = x
H14:I x1
H15:INR x1 = INR x0 - 1
H16:INR x0 = INR x1 + 1
I (Init.Nat.pred x0) /\
(forall i : nat, I i -> (i <= Init.Nat.pred x0)%nat)
  rewrite H15; ring.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:exists i : nat, I i /\ INR i = x
x1:nat
H12:I x1 /\ INR x1 = x
H14:I x1
H15:INR x1 = INR x0 - 1
H16:INR x0 = INR x1 + 1
I (Init.Nat.pred x0) /\
(forall i : nat, I i -> (i <= Init.Nat.pred x0)%nat)
  rewrite <- S_INR in H16; assert (H17 := INR_eq _ _ H16); rewrite H17;
    simpl; split.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:exists i : nat, I i /\ INR i = x
x1:nat
H12:I x1 /\ INR x1 = x
H14:I x1
H15:INR x1 = INR x0 - 1
H16:INR x0 = INR (S x1)
H17:x0 = S x1
I x1
I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:exists i : nat, I i /\ INR i = x
x1:nat
H12:I x1 /\ INR x1 = x
H14:I x1
H15:INR x1 = INR x0 - 1
H16:INR x0 = INR (S x1)
H17:x0 = S x1
forall i : nat, I i -> (i <= x1)%nat
  assumption.I:nat -> Prop
H:exists n : nat, I n
H0:Nbound I
E:=fun x2 : R => exists i : nat, I i /\ INR i = x2:R -> Prop
H1:bound E
H2:exists x2 : R, E x2
x:R
H4:forall x2 : R, E x2 -> x2 <= x
H5:forall b : R,
(forall x2 : R, E x2 -> x2 <= b) -> x <= b
H6:0 <= x
H3:IZR (up x) > x
H7:IZR (up x) - x <= 1
H9:x <= IZR (up x) - 1
H10:x = IZR (up x) - 1
H11:(0 <= up x)%Z
x0:nat
H8:up x = Z.of_nat x0
H13:exists i : nat, I i /\ INR i = x
x1:nat
H12:I x1 /\ INR x1 = x
H14:I x1
H15:INR x1 = INR x0 - 1
H16:INR x0 = INR (S x1)
H17:x0 = S x1
forall i : nat, I i -> (i <= x1)%nat
  intros; apply INR_le; rewrite H15; rewrite <- H15; elim H12; intros;
    rewrite H20; apply H4; unfold E; exists i;
      split; [ assumption | reflexivity ].
Qed.

(*******************************************)

Step functions

(*******************************************)

Definition open_interval (a b x:R) : Prop := a < x < b.
Definition co_interval (a b x:R) : Prop := a <= x < b.

Definition adapted_couple (f:R -> R) (a b:R) (l lf:Rlist) : Prop :=
  ordered_Rlist l /\
  pos_Rl l 0 = Rmin a b /\
  pos_Rl l (pred (Rlength l)) = Rmax a b /\
  Rlength l = S (Rlength lf) /\
  (forall i:nat,
    (i < pred (Rlength l))%nat ->
    constant_D_eq f (open_interval (pos_Rl l i) (pos_Rl l (S i)))
    (pos_Rl lf i)).

Definition adapted_couple_opt (f:R -> R) (a b:R) (l lf:Rlist) :=
  adapted_couple f a b l lf /\
  (forall i:nat,
    (i < pred (Rlength lf))%nat ->
    pos_Rl lf i <> pos_Rl lf (S i) \/ f (pos_Rl l (S i)) <> pos_Rl lf i) /\
  (forall i:nat, (i < pred (Rlength l))%nat -> pos_Rl l i <> pos_Rl l (S i)).

Definition is_subdivision (f:R -> R) (a b:R) (l:Rlist) : Type :=
  { l0:Rlist & adapted_couple f a b l l0 }.

Definition IsStepFun (f:R -> R) (a b:R) : Type :=
  { l:Rlist & is_subdivision f a b l }.

Class of step functions

Record StepFun (a b:R) : Type := mkStepFun
  {fe :> R -> R; pre : IsStepFun fe a b}.

Definition subdivision (a b:R) (f:StepFun a b) : Rlist := projT1 (pre f).

Definition subdivision_val (a b:R) (f:StepFun a b) : Rlist :=
  match projT2 (pre f) with
    | existT _ a b => a
  end.

Fixpoint Int_SF (l k:Rlist) : R :=
  match l with
    | nil => 0
    | cons a l' =>
      match k with
        | nil => 0
        | cons x nil => 0
        | cons x (cons y k') => a * (y - x) + Int_SF l' (cons y k')
      end
  end.

Integral of step functions

Definition RiemannInt_SF (a b:R) (f:StepFun a b) : R :=
  match Rle_dec a b with
    | left _ => Int_SF (subdivision_val f) (subdivision f)
    | right _ => - Int_SF (subdivision_val f) (subdivision f)
  end.

(************************************)

Properties of step functions

(************************************)

Lemma StepFun_P1 :
  forall (a b:R) (f:StepFun a b),
    adapted_couple f a b (subdivision f) (subdivision_val f).forall (a b : R) (f : StepFun a b),
adapted_couple f a b (subdivision f)
  (subdivision_val f)
Proof.forall (a b : R) (f : StepFun a b),
adapted_couple f a b (subdivision f)
  (subdivision_val f)
  intros a b f; unfold subdivision_val; case (projT2 (pre f)) as (x,H);
    apply H.
Qed.

Lemma StepFun_P2 :
  forall (a b:R) (f:R -> R) (l lf:Rlist),
    adapted_couple f a b l lf -> adapted_couple f b a l lf.forall (a b : R) (f : R -> R) (l lf : Rlist),
adapted_couple f a b l lf -> adapted_couple f b a l lf
Proof.forall (a b : R) (f : R -> R) (l lf : Rlist),
adapted_couple f a b l lf -> adapted_couple f b a l lf
  unfold adapted_couple; intros; decompose [and] H; clear H;
    repeat split; try assumption.a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
pos_Rl l 0 = Rmin b a
a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax b a
  rewrite H2; unfold Rmin; case (Rle_dec a b); intro;
    case (Rle_dec b a); intro; try reflexivity.a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
r:a <= b
r0:b <= a
a = b
a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
n:~ a <= b
n0:~ b <= a
b = a
a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax b a
  apply Rle_antisym; assumption.a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
n:~ a <= b
n0:~ b <= a
b = a
a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax b a
  apply Rle_antisym; auto with real.a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax b a
  rewrite H1; unfold Rmax; case (Rle_dec a b); intro;
    case (Rle_dec b a); intro; try reflexivity.a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
r:a <= b
r0:b <= a
b = a
a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
n:~ a <= b
n0:~ b <= a
a = b
  apply Rle_antisym; assumption.a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
n:~ a <= b
n0:~ b <= a
a = b
  apply Rle_antisym; auto with real.
Qed.

Lemma StepFun_P3 :
  forall a b c:R,
    a <= b ->
    adapted_couple (fct_cte c) a b (cons a (cons b nil)) (cons c nil).forall a b c : R,
a <= b ->
adapted_couple (fct_cte c) a b (cons a (cons b nil))
  (cons c nil)
Proof.forall a b c : R,
a <= b ->
adapted_couple (fct_cte c) a b (cons a (cons b nil))
  (cons c nil)
  intros; unfold adapted_couple; repeat split.a, b, c:R
H:a <= b
ordered_Rlist (cons a (cons b nil))
a, b, c:R
H:a <= b
pos_Rl (cons a (cons b nil)) 0 = Rmin a b
a, b, c:R
H:a <= b
pos_Rl (cons a (cons b nil))
  (Init.Nat.pred (Rlength (cons a (cons b nil)))) =
Rmax a b
a, b, c:R
H:a <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
constant_D_eq (fct_cte c)
  (open_interval (pos_Rl (cons a (cons b nil)) i)
     (pos_Rl (cons a (cons b nil)) (S i)))
  (pos_Rl (cons c nil) i)
  unfold ordered_Rlist; intros; simpl in H0; inversion H0;
    [ simpl; assumption | elim (le_Sn_O _ H2) ].a, b, c:R
H:a <= b
pos_Rl (cons a (cons b nil)) 0 = Rmin a b
a, b, c:R
H:a <= b
pos_Rl (cons a (cons b nil))
  (Init.Nat.pred (Rlength (cons a (cons b nil)))) =
Rmax a b
a, b, c:R
H:a <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
constant_D_eq (fct_cte c)
  (open_interval (pos_Rl (cons a (cons b nil)) i)
     (pos_Rl (cons a (cons b nil)) (S i)))
  (pos_Rl (cons c nil) i)
  simpl; unfold Rmin; decide (Rle_dec a b) with H; reflexivity.a, b, c:R
H:a <= b
pos_Rl (cons a (cons b nil))
  (Init.Nat.pred (Rlength (cons a (cons b nil)))) =
Rmax a b
a, b, c:R
H:a <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
constant_D_eq (fct_cte c)
  (open_interval (pos_Rl (cons a (cons b nil)) i)
     (pos_Rl (cons a (cons b nil)) (S i)))
  (pos_Rl (cons c nil) i)
  simpl; unfold Rmax; decide (Rle_dec a b) with H; reflexivity.a, b, c:R
H:a <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
constant_D_eq (fct_cte c)
  (open_interval (pos_Rl (cons a (cons b nil)) i)
     (pos_Rl (cons a (cons b nil)) (S i)))
  (pos_Rl (cons c nil) i)
  unfold constant_D_eq, open_interval; intros; simpl in H0;
    inversion H0; [ reflexivity | elim (le_Sn_O _ H3) ].
Qed.

Lemma StepFun_P4 : forall a b c:R, IsStepFun (fct_cte c) a b.forall a b c : R, IsStepFun (fct_cte c) a b
Proof.forall a b c : R, IsStepFun (fct_cte c) a b
  intros; unfold IsStepFun; destruct (Rle_dec a b) as [Hle|Hnle].a, b, c:R
Hle:a <= b
{l : Rlist & is_subdivision (fct_cte c) a b l}
a, b, c:R
Hnle:~ a <= b
{l : Rlist & is_subdivision (fct_cte c) a b l}
  apply existT with (cons a (cons b nil)); unfold is_subdivision;
    apply existT with (cons c nil); apply (StepFun_P3 c Hle).a, b, c:R
Hnle:~ a <= b
{l : Rlist & is_subdivision (fct_cte c) a b l}
  apply existT with (cons b (cons a nil)); unfold is_subdivision;
    apply existT with (cons c nil); apply StepFun_P2;
      apply StepFun_P3; auto with real.
Qed.

Lemma StepFun_P5 :
  forall (a b:R) (f:R -> R) (l:Rlist),
    is_subdivision f a b l -> is_subdivision f b a l.forall (a b : R) (f : R -> R) (l : Rlist),
is_subdivision f a b l -> is_subdivision f b a l
Proof.forall (a b : R) (f : R -> R) (l : Rlist),
is_subdivision f a b l -> is_subdivision f b a l
  destruct 1 as (x,(H0,(H1,(H2,(H3,H4))))); exists x;
    repeat split; try assumption.a, b:R
f:R -> R
l, x:Rlist
H0:ordered_Rlist l
H1:pos_Rl l 0 = Rmin a b
H2:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength x)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl x i)
pos_Rl l 0 = Rmin b a
a, b:R
f:R -> R
l, x:Rlist
H0:ordered_Rlist l
H1:pos_Rl l 0 = Rmin a b
H2:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength x)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl x i)
pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax b a
  rewrite H1; apply Rmin_comm.a, b:R
f:R -> R
l, x:Rlist
H0:ordered_Rlist l
H1:pos_Rl l 0 = Rmin a b
H2:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength x)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl x i)
pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax b a
  rewrite H2; apply Rmax_comm.
Qed.

Lemma StepFun_P6 :
  forall (f:R -> R) (a b:R), IsStepFun f a b -> IsStepFun f b a.forall (f : R -> R) (a b : R),
IsStepFun f a b -> IsStepFun f b a
Proof.forall (f : R -> R) (a b : R),
IsStepFun f a b -> IsStepFun f b a
  unfold IsStepFun; intros; elim X; intros; apply existT with x;
    apply StepFun_P5; assumption.
Qed.

Lemma StepFun_P7 :
  forall (a b r1 r2 r3:R) (f:R -> R) (l lf:Rlist),
    a <= b ->
    adapted_couple f a b (cons r1 (cons r2 l)) (cons r3 lf) ->
    adapted_couple f r2 b (cons r2 l) lf.forall (a b r1 r2 r3 : R) (f : R -> R) (l lf : Rlist),
a <= b ->
adapted_couple f a b (cons r1 (cons r2 l))
  (cons r3 lf) -> adapted_couple f r2 b (cons r2 l) lf
Proof.forall (a b r1 r2 r3 : R) (f : R -> R) (l lf : Rlist),
a <= b ->
adapted_couple f a b (cons r1 (cons r2 l))
  (cons r3 lf) -> adapted_couple f r2 b (cons r2 l) lf
  unfold adapted_couple; intros; decompose [and] H0; clear H0;
    assert (H5 : Rmax a b = b).a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
Rmax a b = b
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
ordered_Rlist (cons r2 l) /\
pos_Rl (cons r2 l) 0 = Rmin r2 b /\
pos_Rl (cons r2 l)
  (Init.Nat.pred (Rlength (cons r2 l))) = Rmax r2 b /\
Rlength (cons r2 l) = S (Rlength lf) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 l)))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl (cons r2 l) i)
      (pos_Rl (cons r2 l) (S i))) (pos_Rl lf i))
  unfold Rmax; decide (Rle_dec a b) with H; reflexivity.a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
ordered_Rlist (cons r2 l) /\
pos_Rl (cons r2 l) 0 = Rmin r2 b /\
pos_Rl (cons r2 l)
  (Init.Nat.pred (Rlength (cons r2 l))) = Rmax r2 b /\
Rlength (cons r2 l) = S (Rlength lf) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 l)))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl (cons r2 l) i)
      (pos_Rl (cons r2 l) (S i))) (pos_Rl lf i))
  assert (H7 : r2 <= b).a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
r2 <= b
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
ordered_Rlist (cons r2 l) /\
pos_Rl (cons r2 l) 0 = Rmin r2 b /\
pos_Rl (cons r2 l)
  (Init.Nat.pred (Rlength (cons r2 l))) = Rmax r2 b /\
Rlength (cons r2 l) = S (Rlength lf) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 l)))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl (cons r2 l) i)
      (pos_Rl (cons r2 l) (S i))) (pos_Rl lf i))
  rewrite H5 in H2; rewrite <- H2; apply RList_P7;
    [ assumption | simpl; right; left; reflexivity ].a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
ordered_Rlist (cons r2 l) /\
pos_Rl (cons r2 l) 0 = Rmin r2 b /\
pos_Rl (cons r2 l)
  (Init.Nat.pred (Rlength (cons r2 l))) = Rmax r2 b /\
Rlength (cons r2 l) = S (Rlength lf) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 l)))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl (cons r2 l) i)
      (pos_Rl (cons r2 l) (S i))) (pos_Rl lf i))
  repeat split.a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
ordered_Rlist (cons r2 l)
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
pos_Rl (cons r2 l) 0 = Rmin r2 b
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
pos_Rl (cons r2 l)
  (Init.Nat.pred (Rlength (cons r2 l))) = Rmax r2 b
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
Rlength (cons r2 l) = S (Rlength lf)
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r2 l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r2 l) i)
     (pos_Rl (cons r2 l) (S i))) (pos_Rl lf i)
  apply RList_P4 with r1; assumption.a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
pos_Rl (cons r2 l) 0 = Rmin r2 b
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
pos_Rl (cons r2 l)
  (Init.Nat.pred (Rlength (cons r2 l))) = Rmax r2 b
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
Rlength (cons r2 l) = S (Rlength lf)
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r2 l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r2 l) i)
     (pos_Rl (cons r2 l) (S i))) (pos_Rl lf i)
  rewrite H5 in H2; unfold Rmin; decide (Rle_dec r2 b) with H7; reflexivity.a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
pos_Rl (cons r2 l)
  (Init.Nat.pred (Rlength (cons r2 l))) = Rmax r2 b
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
Rlength (cons r2 l) = S (Rlength lf)
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r2 l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r2 l) i)
     (pos_Rl (cons r2 l) (S i))) (pos_Rl lf i)
  unfold Rmax; decide (Rle_dec r2 b) with H7.a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
pos_Rl (cons r2 l)
  (Init.Nat.pred (Rlength (cons r2 l))) = b
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
Rlength (cons r2 l) = S (Rlength lf)
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r2 l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r2 l) i)
     (pos_Rl (cons r2 l) (S i))) (pos_Rl lf i)
    rewrite H5 in H2; rewrite <- H2; reflexivity.a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
Rlength (cons r2 l) = S (Rlength lf)
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r2 l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r2 l) i)
     (pos_Rl (cons r2 l) (S i))) (pos_Rl lf i)
  simpl in H4; simpl; apply INR_eq; apply Rplus_eq_reg_l with 1;
    do 2 rewrite (Rplus_comm 1); do 2 rewrite <- S_INR;
      rewrite H4; reflexivity.a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r1 (cons r2 l)) i)
     (pos_Rl (cons r1 (cons r2 l)) (S i)))
  (pos_Rl (cons r3 lf) i)
H5:Rmax a b = b
H7:r2 <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r2 l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r2 l) i)
     (pos_Rl (cons r2 l) (S i))) (pos_Rl lf i)
  intros; unfold constant_D_eq, open_interval; intros;
    unfold constant_D_eq, open_interval in H6;
      assert (H9 : (S i < pred (Rlength (cons r1 (cons r2 l))))%nat).a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
forall x0 : R,
pos_Rl (cons r1 (cons r2 l)) i0 < x0 <
pos_Rl (cons r1 (cons r2 l)) (S i0) ->
f x0 = pos_Rl (cons r3 lf) i0
H5:Rmax a b = b
H7:r2 <= b
i:nat
H0:(i < Init.Nat.pred (Rlength (cons r2 l)))%nat
x:R
H8:pos_Rl (cons r2 l) i < x <
pos_Rl (cons r2 l) (S i)
(S i < Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat
a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
forall x0 : R,
pos_Rl (cons r1 (cons r2 l)) i0 < x0 <
pos_Rl (cons r1 (cons r2 l)) (S i0) ->
f x0 = pos_Rl (cons r3 lf) i0
H5:Rmax a b = b
H7:r2 <= b
i:nat
H0:(i < Init.Nat.pred (Rlength (cons r2 l)))%nat
x:R
H8:pos_Rl (cons r2 l) i < x <
pos_Rl (cons r2 l) (S i)
H9:(S i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat
f x = pos_Rl lf i
  simpl; simpl in H0; apply lt_n_S; assumption.a, b, r1, r2, r3:R
f:R -> R
l, lf:Rlist
H:a <= b
H1:ordered_Rlist (cons r1 (cons r2 l))
H3:pos_Rl (cons r1 (cons r2 l)) 0 = Rmin a b
H2:pos_Rl (cons r1 (cons r2 l))
  (Init.Nat.pred (Rlength (cons r1 (cons r2 l)))) =
Rmax a b
H4:Rlength (cons r1 (cons r2 l)) =
S (Rlength (cons r3 lf))
H6:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat ->
forall x0 : R,
pos_Rl (cons r1 (cons r2 l)) i0 < x0 <
pos_Rl (cons r1 (cons r2 l)) (S i0) ->
f x0 = pos_Rl (cons r3 lf) i0
H5:Rmax a b = b
H7:r2 <= b
i:nat
H0:(i < Init.Nat.pred (Rlength (cons r2 l)))%nat
x:R
H8:pos_Rl (cons r2 l) i < x <
pos_Rl (cons r2 l) (S i)
H9:(S i <
 Init.Nat.pred (Rlength (cons r1 (cons r2 l))))%nat
f x = pos_Rl lf i
  assert (H10 := H6 _ H9); apply H10; assumption.
Qed.

Lemma StepFun_P8 :
  forall (f:R -> R) (l1 lf1:Rlist) (a b:R),
    adapted_couple f a b l1 lf1 -> a = b -> Int_SF lf1 l1 = 0.forall (f : R -> R) (l1 lf1 : Rlist) (a b : R),
adapted_couple f a b l1 lf1 ->
a = b -> Int_SF lf1 l1 = 0
Proof.forall (f : R -> R) (l1 lf1 : Rlist) (a b : R),
adapted_couple f a b l1 lf1 ->
a = b -> Int_SF lf1 l1 = 0
  simple induction l1.f:R -> R
l1:Rlist
forall (lf1 : Rlist) (a b : R),
adapted_couple f a b nil lf1 ->
a = b -> Int_SF lf1 nil = 0
f:R -> R
l1:Rlist
forall (r : R) (r0 : Rlist),
(forall (lf1 : Rlist) (a b : R),
 adapted_couple f a b r0 lf1 ->
 a = b -> Int_SF lf1 r0 = 0) ->
forall (lf1 : Rlist) (a b : R),
adapted_couple f a b (cons r r0) lf1 ->
a = b -> Int_SF lf1 (cons r r0) = 0
  intros; induction  lf1 as [| r lf1 Hreclf1]; reflexivity.f:R -> R
l1:Rlist
forall (r : R) (r0 : Rlist),
(forall (lf1 : Rlist) (a b : R),
 adapted_couple f a b r0 lf1 ->
 a = b -> Int_SF lf1 r0 = 0) ->
forall (lf1 : Rlist) (a b : R),
adapted_couple f a b (cons r r0) lf1 ->
a = b -> Int_SF lf1 (cons r r0) = 0
  simple induction r0.f:R -> R
l1:Rlist
r:R
r0:Rlist
(forall (lf1 : Rlist) (a b : R),
 adapted_couple f a b nil lf1 ->
 a = b -> Int_SF lf1 nil = 0) ->
forall (lf1 : Rlist) (a b : R),
adapted_couple f a b (cons r nil) lf1 ->
a = b -> Int_SF lf1 (cons r nil) = 0
f:R -> R
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b : R),
  adapted_couple f a b r2 lf1 ->
  a = b -> Int_SF lf1 r2 = 0) ->
 forall (lf1 : Rlist) (a b : R),
 adapted_couple f a b (cons r r2) lf1 ->
 a = b -> Int_SF lf1 (cons r r2) = 0) ->
(forall (lf1 : Rlist) (a b : R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a = b -> Int_SF lf1 (cons r1 r2) = 0) ->
forall (lf1 : Rlist) (a b : R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a = b -> Int_SF lf1 (cons r (cons r1 r2)) = 0
  intros; induction  lf1 as [| r1 lf1 Hreclf1].f:R -> R
l1:Rlist
r:R
r0:Rlist
H:forall (lf1 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 nil lf1 ->
a0 = b0 -> Int_SF lf1 nil = 0
a, b:R
H0:adapted_couple f a b (cons r nil) nil
H1:a = b
Int_SF nil (cons r nil) = 0
f:R -> R
l1:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 nil lf0 ->
a0 = b0 -> Int_SF lf0 nil = 0
r1:R
lf1:Rlist
a, b:R
H0:adapted_couple f a b (cons r nil) (cons r1 lf1)
H1:a = b
Hreclf1:adapted_couple f a b (cons r nil) lf1 ->
Int_SF lf1 (cons r nil) = 0
Int_SF (cons r1 lf1) (cons r nil) = 0
f:R -> R
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b : R),
  adapted_couple f a b r2 lf1 ->
  a = b -> Int_SF lf1 r2 = 0) ->
 forall (lf1 : Rlist) (a b : R),
 adapted_couple f a b (cons r r2) lf1 ->
 a = b -> Int_SF lf1 (cons r r2) = 0) ->
(forall (lf1 : Rlist) (a b : R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a = b -> Int_SF lf1 (cons r1 r2) = 0) ->
forall (lf1 : Rlist) (a b : R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a = b -> Int_SF lf1 (cons r (cons r1 r2)) = 0
  reflexivity.f:R -> R
l1:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 nil lf0 ->
a0 = b0 -> Int_SF lf0 nil = 0
r1:R
lf1:Rlist
a, b:R
H0:adapted_couple f a b (cons r nil) (cons r1 lf1)
H1:a = b
Hreclf1:adapted_couple f a b (cons r nil) lf1 ->
Int_SF lf1 (cons r nil) = 0
Int_SF (cons r1 lf1) (cons r nil) = 0
f:R -> R
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b : R),
  adapted_couple f a b r2 lf1 ->
  a = b -> Int_SF lf1 r2 = 0) ->
 forall (lf1 : Rlist) (a b : R),
 adapted_couple f a b (cons r r2) lf1 ->
 a = b -> Int_SF lf1 (cons r r2) = 0) ->
(forall (lf1 : Rlist) (a b : R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a = b -> Int_SF lf1 (cons r1 r2) = 0) ->
forall (lf1 : Rlist) (a b : R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a = b -> Int_SF lf1 (cons r (cons r1 r2)) = 0
  unfold adapted_couple in H0; decompose [and] H0; clear H0; simpl in H5;
    discriminate.f:R -> R
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b : R),
  adapted_couple f a b r2 lf1 ->
  a = b -> Int_SF lf1 r2 = 0) ->
 forall (lf1 : Rlist) (a b : R),
 adapted_couple f a b (cons r r2) lf1 ->
 a = b -> Int_SF lf1 (cons r r2) = 0) ->
(forall (lf1 : Rlist) (a b : R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a = b -> Int_SF lf1 (cons r1 r2) = 0) ->
forall (lf1 : Rlist) (a b : R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a = b -> Int_SF lf1 (cons r (cons r1 r2)) = 0
  intros; induction  lf1 as [| r3 lf1 Hreclf1].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf1 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf1 ->
 a0 = b0 -> Int_SF lf1 r2 = 0) ->
forall (lf1 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf1 ->
a0 = b0 -> Int_SF lf1 (cons r r2) = 0
H0:forall (lf1 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf1 ->
a0 = b0 -> Int_SF lf1 (cons r1 r2) = 0
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2)) nil
H2:a = b
Int_SF nil (cons r (cons r1 r2)) = 0
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) = 0
  reflexivity.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) = 0
  simpl; cut (r = r1).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
r = r1 -> r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) = 0
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
r = r1
  intro; rewrite H3; rewrite (H0 lf1 r b).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
H3:r = r1
r3 * (r1 - r1) + 0 = 0
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
H3:r = r1
adapted_couple f r b (cons r1 r2) lf1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
H3:r = r1
r = b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
r = r1
  ring.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
H3:r = r1
adapted_couple f r b (cons r1 r2) lf1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
H3:r = r1
r = b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
r = r1
  rewrite H3; apply StepFun_P7 with a r r3; [ right; assumption | assumption ].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
H3:r = r1
r = b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
r = r1
  clear H H0 Hreclf1 r0; unfold adapted_couple in H1; decompose [and] H1;
    intros; simpl in H4; rewrite H4; unfold Rmin;
      case (Rle_dec a b); intro; [ assumption | reflexivity ].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
r = r1
  unfold adapted_couple in H1; decompose [and] H1; intros; apply Rle_antisym.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:ordered_Rlist (cons r (cons r1 r2)) /\
pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b /\
pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b /\
Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1)) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
 constant_D_eq f
   (open_interval
      (pos_Rl (cons r (cons r1 r2)) i)
      (pos_Rl (cons r (cons r1 r2)) (S i)))
   (pos_Rl (cons r3 lf1) i))
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
r <= r1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:ordered_Rlist (cons r (cons r1 r2)) /\
pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b /\
pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b /\
Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1)) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
 constant_D_eq f
   (open_interval
      (pos_Rl (cons r (cons r1 r2)) i)
      (pos_Rl (cons r (cons r1 r2)) (S i)))
   (pos_Rl (cons r3 lf1) i))
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
r1 <= r
  apply (H3 0%nat); simpl; apply lt_O_Sn.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H:(forall (lf2 : Rlist) (a0 b0 : R),
 adapted_couple f a0 b0 r2 lf2 ->
 a0 = b0 -> Int_SF lf2 r2 = 0) ->
forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r r2) = 0
H0:forall (lf0 : Rlist) (a0 b0 : R),
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
a0 = b0 -> Int_SF lf0 (cons r1 r2) = 0
r3:R
lf1:Rlist
a, b:R
H1:ordered_Rlist (cons r (cons r1 r2)) /\
pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b /\
pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b /\
Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1)) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
 constant_D_eq f
   (open_interval
      (pos_Rl (cons r (cons r1 r2)) i)
      (pos_Rl (cons r (cons r1 r2)) (S i)))
   (pos_Rl (cons r3 lf1) i))
H2:a = b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) = 0
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
r1 <= r
  simpl in H5; rewrite H2 in H5; rewrite H5; replace (Rmin b b) with (Rmax a b);
    [ rewrite <- H4; apply RList_P7;
      [ assumption | simpl; right; left; reflexivity ]
      | unfold Rmin, Rmax; case (Rle_dec b b); case (Rle_dec a b); intros;
        try assumption || reflexivity ].
Qed.

Lemma StepFun_P9 :
  forall (a b:R) (f:R -> R) (l lf:Rlist),
    adapted_couple f a b l lf -> a <> b -> (2 <= Rlength l)%nat.forall (a b : R) (f : R -> R) (l lf : Rlist),
adapted_couple f a b l lf ->
a <> b -> (2 <= Rlength l)%nat
Proof.forall (a b : R) (f : R -> R) (l lf : Rlist),
adapted_couple f a b l lf ->
a <> b -> (2 <= Rlength l)%nat
  intros; unfold adapted_couple in H; decompose [and] H; clear H;
    induction  l as [| r l Hrecl];
      [ simpl in H4; discriminate
        | induction  l as [| r0 l Hrecl0];
          [ simpl in H3; simpl in H2; generalize H3; generalize H2;
            unfold Rmin, Rmax; case (Rle_dec a b);
              intros; elim H0; rewrite <- H5; rewrite <- H7;
                reflexivity
            | simpl; do 2 apply le_n_S; apply le_O_n ] ].
Qed.

Lemma StepFun_P10 :
  forall (f:R -> R) (l lf:Rlist) (a b:R),
    a <= b ->
    adapted_couple f a b l lf ->
    exists l' : Rlist,
      (exists lf' : Rlist, adapted_couple_opt f a b l' lf').forall (f : R -> R) (l lf : Rlist) (a b : R),
a <= b ->
adapted_couple f a b l lf ->
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
Proof.forall (f : R -> R) (l lf : Rlist) (a b : R),
a <= b ->
adapted_couple f a b l lf ->
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
  simple induction l.f:R -> R
l:Rlist
forall (lf : Rlist) (a b : R),
a <= b ->
adapted_couple f a b nil lf ->
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
f:R -> R
l:Rlist
forall (r : R) (r0 : Rlist),
(forall (lf : Rlist) (a b : R),
 a <= b ->
 adapted_couple f a b r0 lf ->
 exists l' lf' : Rlist,
   adapted_couple_opt f a b l' lf') ->
forall (lf : Rlist) (a b : R),
a <= b ->
adapted_couple f a b (cons r r0) lf ->
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
  intros; unfold adapted_couple in H0; decompose [and] H0; simpl in H4;
    discriminate.f:R -> R
l:Rlist
forall (r : R) (r0 : Rlist),
(forall (lf : Rlist) (a b : R),
 a <= b ->
 adapted_couple f a b r0 lf ->
 exists l' lf' : Rlist,
   adapted_couple_opt f a b l' lf') ->
forall (lf : Rlist) (a b : R),
a <= b ->
adapted_couple f a b (cons r r0) lf ->
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
  intros; case (Req_dec a b); intro.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) lf
H2:a = b
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) lf
H2:a <> b
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
  exists (cons a nil); exists nil; unfold adapted_couple_opt;
    unfold adapted_couple; unfold ordered_Rlist;
      repeat split; try (intros; simpl in H3; elim (lt_n_O _ H3)).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) lf
H2:a = b
pos_Rl (cons a nil) 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) lf
H2:a = b
pos_Rl (cons a nil)
  (Init.Nat.pred (Rlength (cons a nil))) = Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) lf
H2:a <> b
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
  simpl; rewrite <- H2; unfold Rmin; case (Rle_dec a a); intro;
    reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) lf
H2:a = b
pos_Rl (cons a nil)
  (Init.Nat.pred (Rlength (cons a nil))) = Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) lf
H2:a <> b
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
  simpl; rewrite <- H2; unfold Rmax; case (Rle_dec a a); intro;
    reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) lf
H2:a <> b
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
  elim (RList_P20 _ (StepFun_P9 H1 H2)); intros t1 [t2 [t3 H3]];
    induction  lf as [| r1 lf Hreclf].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) nil
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
Hreclf:adapted_couple f a b (cons r r0) lf ->
exists l' lf' : Rlist,
  adapted_couple_opt f a b l' lf'
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
  unfold adapted_couple in H1; decompose [and] H1; rewrite H3 in H7;
    simpl in H7; discriminate.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
Hreclf:adapted_couple f a b (cons r r0) lf ->
exists l' lf' : Rlist,
  adapted_couple_opt f a b l' lf'
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
  clear Hreclf; assert (H4 : adapted_couple f t2 b r0 lf).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
adapted_couple f t2 b r0 lf
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
  rewrite H3 in H1; assert (H4 := RList_P21 _ _ H3); simpl in H4; rewrite H4;
    eapply StepFun_P7; [ apply H0 | apply H1 ].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
  cut (t2 <= b).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b ->
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  intro; assert (H6 := H _ _ _ H5 H4); case (Req_dec t1 t2); intro Hyp_eq.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
H6:exists l' lf' : Rlist,
  adapted_couple_opt f t2 b l' lf'
Hyp_eq:t1 = t2
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
H6:exists l' lf' : Rlist,
  adapted_couple_opt f t2 b l' lf'
Hyp_eq:t1 <> t2
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  replace a with t2.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
H6:exists l' lf' : Rlist,
  adapted_couple_opt f t2 b l' lf'
Hyp_eq:t1 = t2
exists l' lf' : Rlist,
  adapted_couple_opt f t2 b l' lf'
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
H6:exists l' lf' : Rlist,
  adapted_couple_opt f t2 b l' lf'
Hyp_eq:t1 = t2
t2 = a
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
H6:exists l' lf' : Rlist,
  adapted_couple_opt f t2 b l' lf'
Hyp_eq:t1 <> t2
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  apply H6.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
H6:exists l' lf' : Rlist,
  adapted_couple_opt f t2 b l' lf'
Hyp_eq:t1 = t2
t2 = a
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
H6:exists l' lf' : Rlist,
  adapted_couple_opt f t2 b l' lf'
Hyp_eq:t1 <> t2
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite <- Hyp_eq; rewrite H3 in H1; unfold adapted_couple in H1;
    decompose [and] H1; clear H1; simpl in H9; rewrite H9;
      unfold Rmin; decide (Rle_dec a b) with H0; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
H6:exists l' lf' : Rlist,
  adapted_couple_opt f t2 b l' lf'
Hyp_eq:t1 <> t2
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  elim H6; clear H6; intros l' [lf' H6]; case (Req_dec t2 b); intro.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  exists (cons a (cons b nil)); exists (cons r1 nil);
    unfold adapted_couple_opt; unfold adapted_couple;
      repeat split.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
ordered_Rlist (cons a (cons b nil))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
pos_Rl (cons a (cons b nil)) 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
pos_Rl (cons a (cons b nil))
  (Init.Nat.pred (Rlength (cons a (cons b nil)))) =
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons b nil)) i)
     (pos_Rl (cons a (cons b nil)) (S i)))
  (pos_Rl (cons r1 nil) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 nil)))%nat ->
pos_Rl (cons r1 nil) i <> pos_Rl (cons r1 nil) (S i) \/
f (pos_Rl (cons a (cons b nil)) (S i)) <>
pos_Rl (cons r1 nil) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
pos_Rl (cons a (cons b nil)) i <>
pos_Rl (cons a (cons b nil)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  unfold ordered_Rlist; intros; simpl in H8; inversion H8;
    [ simpl; assumption | elim (le_Sn_O _ H10) ].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
pos_Rl (cons a (cons b nil)) 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
pos_Rl (cons a (cons b nil))
  (Init.Nat.pred (Rlength (cons a (cons b nil)))) =
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons b nil)) i)
     (pos_Rl (cons a (cons b nil)) (S i)))
  (pos_Rl (cons r1 nil) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 nil)))%nat ->
pos_Rl (cons r1 nil) i <> pos_Rl (cons r1 nil) (S i) \/
f (pos_Rl (cons a (cons b nil)) (S i)) <>
pos_Rl (cons r1 nil) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
pos_Rl (cons a (cons b nil)) i <>
pos_Rl (cons a (cons b nil)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; unfold Rmin; decide (Rle_dec a b) with H0; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
pos_Rl (cons a (cons b nil))
  (Init.Nat.pred (Rlength (cons a (cons b nil)))) =
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons b nil)) i)
     (pos_Rl (cons a (cons b nil)) (S i)))
  (pos_Rl (cons r1 nil) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 nil)))%nat ->
pos_Rl (cons r1 nil) i <> pos_Rl (cons r1 nil) (S i) \/
f (pos_Rl (cons a (cons b nil)) (S i)) <>
pos_Rl (cons r1 nil) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
pos_Rl (cons a (cons b nil)) i <>
pos_Rl (cons a (cons b nil)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; unfold Rmax; decide (Rle_dec a b) with H0; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons b nil)) i)
     (pos_Rl (cons a (cons b nil)) (S i)))
  (pos_Rl (cons r1 nil) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 nil)))%nat ->
pos_Rl (cons r1 nil) i <> pos_Rl (cons r1 nil) (S i) \/
f (pos_Rl (cons a (cons b nil)) (S i)) <>
pos_Rl (cons r1 nil) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
pos_Rl (cons a (cons b nil)) i <>
pos_Rl (cons a (cons b nil)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  intros; simpl in H8; inversion H8.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
i:nat
H8:(i < 1)%nat
H10:i = 0%nat
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons b nil)) 0)
     (pos_Rl (cons a (cons b nil)) 1))
  (pos_Rl (cons r1 nil) 0)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
i:nat
H8:(i < 1)%nat
m:nat
H10:(S i <= 0)%nat
H9:m = 0%nat
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons b nil)) i)
     (pos_Rl (cons a (cons b nil)) (S i)))
  (pos_Rl (cons r1 nil) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 nil)))%nat ->
pos_Rl (cons r1 nil) i <> pos_Rl (cons r1 nil) (S i) \/
f (pos_Rl (cons a (cons b nil)) (S i)) <>
pos_Rl (cons r1 nil) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
pos_Rl (cons a (cons b nil)) i <>
pos_Rl (cons a (cons b nil)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  unfold constant_D_eq, open_interval; intros; simpl;
    simpl in H9; rewrite H3 in H1; unfold adapted_couple in H1;
      decompose [and] H1; apply (H16 0%nat).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H1:ordered_Rlist (cons t1 (cons t2 t3)) /\
pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b /\
pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b /\
Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf)) /\
(forall i0 : nat,
 (i0 <
  Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
 constant_D_eq f
   (open_interval
      (pos_Rl (cons t1 (cons t2 t3)) i0)
      (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
   (pos_Rl (cons r1 lf) i0))
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
i:nat
H8:(i < 1)%nat
H10:i = 0%nat
x:R
H9:a < x < b
H11:ordered_Rlist (cons t1 (cons t2 t3))
H13:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H12:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H14:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
(0 < Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H1:ordered_Rlist (cons t1 (cons t2 t3)) /\
pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b /\
pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b /\
Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf)) /\
(forall i0 : nat,
 (i0 <
  Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
 constant_D_eq f
   (open_interval
      (pos_Rl (cons t1 (cons t2 t3)) i0)
      (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
   (pos_Rl (cons r1 lf) i0))
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
i:nat
H8:(i < 1)%nat
H10:i = 0%nat
x:R
H9:a < x < b
H11:ordered_Rlist (cons t1 (cons t2 t3))
H13:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H12:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H14:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
open_interval (pos_Rl (cons t1 (cons t2 t3)) 0)
  (pos_Rl (cons t1 (cons t2 t3)) 1) x
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
i:nat
H8:(i < 1)%nat
m:nat
H10:(S i <= 0)%nat
H9:m = 0%nat
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons b nil)) i)
     (pos_Rl (cons a (cons b nil)) (S i)))
  (pos_Rl (cons r1 nil) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 nil)))%nat ->
pos_Rl (cons r1 nil) i <> pos_Rl (cons r1 nil) (S i) \/
f (pos_Rl (cons a (cons b nil)) (S i)) <>
pos_Rl (cons r1 nil) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
pos_Rl (cons a (cons b nil)) i <>
pos_Rl (cons a (cons b nil)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; apply lt_O_Sn.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H1:ordered_Rlist (cons t1 (cons t2 t3)) /\
pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b /\
pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b /\
Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf)) /\
(forall i0 : nat,
 (i0 <
  Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
 constant_D_eq f
   (open_interval
      (pos_Rl (cons t1 (cons t2 t3)) i0)
      (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
   (pos_Rl (cons r1 lf) i0))
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
i:nat
H8:(i < 1)%nat
H10:i = 0%nat
x:R
H9:a < x < b
H11:ordered_Rlist (cons t1 (cons t2 t3))
H13:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H12:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H14:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
open_interval (pos_Rl (cons t1 (cons t2 t3)) 0)
  (pos_Rl (cons t1 (cons t2 t3)) 1) x
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
i:nat
H8:(i < 1)%nat
m:nat
H10:(S i <= 0)%nat
H9:m = 0%nat
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons b nil)) i)
     (pos_Rl (cons a (cons b nil)) (S i)))
  (pos_Rl (cons r1 nil) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 nil)))%nat ->
pos_Rl (cons r1 nil) i <> pos_Rl (cons r1 nil) (S i) \/
f (pos_Rl (cons a (cons b nil)) (S i)) <>
pos_Rl (cons r1 nil) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
pos_Rl (cons a (cons b nil)) i <>
pos_Rl (cons a (cons b nil)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  unfold open_interval; simpl; rewrite H7; simpl in H13;
    rewrite H13; unfold Rmin; decide (Rle_dec a b) with H0; assumption.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
i:nat
H8:(i < 1)%nat
m:nat
H10:(S i <= 0)%nat
H9:m = 0%nat
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons b nil)) i)
     (pos_Rl (cons a (cons b nil)) (S i)))
  (pos_Rl (cons r1 nil) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 nil)))%nat ->
pos_Rl (cons r1 nil) i <> pos_Rl (cons r1 nil) (S i) \/
f (pos_Rl (cons a (cons b nil)) (S i)) <>
pos_Rl (cons r1 nil) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
pos_Rl (cons a (cons b nil)) i <>
pos_Rl (cons a (cons b nil)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  elim (le_Sn_O _ H10).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 nil)))%nat ->
pos_Rl (cons r1 nil) i <> pos_Rl (cons r1 nil) (S i) \/
f (pos_Rl (cons a (cons b nil)) (S i)) <>
pos_Rl (cons r1 nil) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
pos_Rl (cons a (cons b nil)) i <>
pos_Rl (cons a (cons b nil)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  intros; simpl in H8; elim (lt_n_O _ H8).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons b nil))))%nat ->
pos_Rl (cons a (cons b nil)) i <>
pos_Rl (cons a (cons b nil)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  intros; simpl in H8; inversion H8;
    [ simpl; assumption | elim (le_Sn_O _ H10) ].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  assert (Hyp_min : Rmin t2 b = t2).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
Rmin t2 b = t2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  unfold Rmin; decide (Rle_dec t2 b) with H5; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l', lf':Rlist
H6:adapted_couple_opt f t2 b l' lf'
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  unfold adapted_couple in H6; elim H6; clear H6; intros;
    elim (RList_P20 _ (StepFun_P9 H6 H7)); intros s1 [s2 [s3 H9]];
      induction  lf' as [| r2 lf' Hreclf'].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf' : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' nil
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength nil))%nat ->
 pos_Rl nil i <> pos_Rl nil (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl nil i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
exists l'0 lf' : Rlist,
  adapted_couple_opt f a b l'0 lf'
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
Hreclf':adapted_couple f t2 b l' lf' ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf'))%nat ->
 pos_Rl lf' i <> pos_Rl lf' (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl lf' i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i)) ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  unfold adapted_couple in H6; decompose [and] H6; rewrite H9 in H13;
    simpl in H13; discriminate.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
Hreclf':adapted_couple f t2 b l' lf' ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf'))%nat ->
 pos_Rl lf' i <> pos_Rl lf' (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl lf' i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i)) ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  clear Hreclf'; case (Req_dec r1 r2); intro.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  case (Req_dec (f t2) r1); intro.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  exists (cons t1 (cons s2 s3)); exists (cons r1 lf'); rewrite H3 in H1;
    rewrite H9 in H6; unfold adapted_couple in H6, H1;
      decompose [and] H1; decompose [and] H6; clear H1 H6;
        unfold adapted_couple_opt; unfold adapted_couple;
          repeat split.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
ordered_Rlist (cons t1 (cons s2 s3))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3)) 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons t1 (cons s2 s3)))) =
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
Rlength (cons t1 (cons s2 s3)) =
S (Rlength (cons r1 lf'))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 (cons s2 s3)) i)
     (pos_Rl (cons t1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r1 lf') i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  unfold ordered_Rlist; intros; simpl in H1;
    induction  i as [| i Hreci].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
pos_Rl (cons t1 (cons s2 s3)) 0 <=
pos_Rl (cons t1 (cons s2 s3)) 1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i < S (Rlength s3))%nat
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <=
pos_Rl (cons t1 (cons s2 s3)) (S i)
pos_Rl (cons t1 (cons s2 s3)) (S i) <=
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3)) 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons t1 (cons s2 s3)))) =
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
Rlength (cons t1 (cons s2 s3)) =
S (Rlength (cons r1 lf'))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 (cons s2 s3)) i)
     (pos_Rl (cons t1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r1 lf') i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; apply Rle_trans with s1.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
t1 <= s1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
s1 <= s2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i < S (Rlength s3))%nat
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <=
pos_Rl (cons t1 (cons s2 s3)) (S i)
pos_Rl (cons t1 (cons s2 s3)) (S i) <=
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3)) 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons t1 (cons s2 s3)))) =
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
Rlength (cons t1 (cons s2 s3)) =
S (Rlength (cons r1 lf'))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 (cons s2 s3)) i)
     (pos_Rl (cons t1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r1 lf') i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  replace s1 with t2.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
t1 <= t2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
t2 = s1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
s1 <= s2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i < S (Rlength s3))%nat
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <=
pos_Rl (cons t1 (cons s2 s3)) (S i)
pos_Rl (cons t1 (cons s2 s3)) (S i) <=
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3)) 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons t1 (cons s2 s3)))) =
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
Rlength (cons t1 (cons s2 s3)) =
S (Rlength (cons r1 lf'))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 (cons s2 s3)) i)
     (pos_Rl (cons t1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r1 lf') i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  apply (H12 0%nat).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
(0 < Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
t2 = s1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
s1 <= s2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i < S (Rlength s3))%nat
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <=
pos_Rl (cons t1 (cons s2 s3)) (S i)
pos_Rl (cons t1 (cons s2 s3)) (S i) <=
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3)) 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons t1 (cons s2 s3)))) =
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
Rlength (cons t1 (cons s2 s3)) =
S (Rlength (cons r1 lf'))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 (cons s2 s3)) i)
     (pos_Rl (cons t1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r1 lf') i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; apply lt_O_Sn.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
t2 = s1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
s1 <= s2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i < S (Rlength s3))%nat
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <=
pos_Rl (cons t1 (cons s2 s3)) (S i)
pos_Rl (cons t1 (cons s2 s3)) (S i) <=
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3)) 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons t1 (cons s2 s3)))) =
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
Rlength (cons t1 (cons s2 s3)) =
S (Rlength (cons r1 lf'))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 (cons s2 s3)) i)
     (pos_Rl (cons t1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r1 lf') i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl in H19; rewrite H19; symmetry ; apply Hyp_min.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
s1 <= s2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i < S (Rlength s3))%nat
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <=
pos_Rl (cons t1 (cons s2 s3)) (S i)
pos_Rl (cons t1 (cons s2 s3)) (S i) <=
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3)) 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons t1 (cons s2 s3)))) =
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
Rlength (cons t1 (cons s2 s3)) =
S (Rlength (cons r1 lf'))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 (cons s2 s3)) i)
     (pos_Rl (cons t1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r1 lf') i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  apply (H16 0%nat); simpl; apply lt_O_Sn.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i < S (Rlength s3))%nat
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <=
pos_Rl (cons t1 (cons s2 s3)) (S i)
pos_Rl (cons t1 (cons s2 s3)) (S i) <=
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3)) 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons t1 (cons s2 s3)))) =
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
Rlength (cons t1 (cons s2 s3)) =
S (Rlength (cons r1 lf'))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 (cons s2 s3)) i)
     (pos_Rl (cons t1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r1 lf') i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  change (pos_Rl (cons s2 s3) i <= pos_Rl (cons s2 s3) (S i));
    apply (H16 (S i)); simpl; assumption.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3)) 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons t1 (cons s2 s3)))) =
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
Rlength (cons t1 (cons s2 s3)) =
S (Rlength (cons r1 lf'))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 (cons s2 s3)) i)
     (pos_Rl (cons t1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r1 lf') i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; simpl in H14; rewrite H14; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
pos_Rl (cons t1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons t1 (cons s2 s3)))) =
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
Rlength (cons t1 (cons s2 s3)) =
S (Rlength (cons r1 lf'))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 (cons s2 s3)) i)
     (pos_Rl (cons t1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r1 lf') i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; simpl in H18; rewrite H18; unfold Rmax;
    decide (Rle_dec a b) with H0; decide (Rle_dec t2 b) with H5; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
Rlength (cons t1 (cons s2 s3)) =
S (Rlength (cons r1 lf'))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 (cons s2 s3)) i)
     (pos_Rl (cons t1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r1 lf') i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; simpl in H20; apply H20.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 (cons s2 s3)) i)
     (pos_Rl (cons t1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r1 lf') i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  intros; simpl in H1; unfold constant_D_eq, open_interval; intros;
    induction  i as [| i Hreci].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
x:R
H6:pos_Rl (cons t1 (cons s2 s3)) 0 < x <
pos_Rl (cons t1 (cons s2 s3)) 1
f x = pos_Rl (cons r1 lf') 0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i < S (Rlength s3))%nat
x:R
H6:pos_Rl (cons t1 (cons s2 s3)) (S i) < x <
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i < x <
pos_Rl (cons t1 (cons s2 s3)) (S i) ->
f x = pos_Rl (cons r1 lf') i
f x = pos_Rl (cons r1 lf') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; simpl in H6; destruct (total_order_T x t2) as [[Hlt|Heq]|Hgt].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
x:R
H6:t1 < x < s2
Hlt:x < t2
f x = r1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
x:R
H6:t1 < x < s2
Heq:x = t2
f x = r1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
x:R
H6:t1 < x < s2
Hgt:x > t2
f x = r1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i < S (Rlength s3))%nat
x:R
H6:pos_Rl (cons t1 (cons s2 s3)) (S i) < x <
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i < x <
pos_Rl (cons t1 (cons s2 s3)) (S i) ->
f x = pos_Rl (cons r1 lf') i
f x = pos_Rl (cons r1 lf') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  apply (H17 0%nat);
    [ simpl; apply lt_O_Sn
      | unfold open_interval; simpl; elim H6; intros; split;
        assumption ].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
x:R
H6:t1 < x < s2
Heq:x = t2
f x = r1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
x:R
H6:t1 < x < s2
Hgt:x > t2
f x = r1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i < S (Rlength s3))%nat
x:R
H6:pos_Rl (cons t1 (cons s2 s3)) (S i) < x <
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i < x <
pos_Rl (cons t1 (cons s2 s3)) (S i) ->
f x = pos_Rl (cons r1 lf') i
f x = pos_Rl (cons r1 lf') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite Heq; assumption.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 < S (Rlength s3))%nat
x:R
H6:t1 < x < s2
Hgt:x > t2
f x = r1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i < S (Rlength s3))%nat
x:R
H6:pos_Rl (cons t1 (cons s2 s3)) (S i) < x <
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i < x <
pos_Rl (cons t1 (cons s2 s3)) (S i) ->
f x = pos_Rl (cons r1 lf') i
f x = pos_Rl (cons r1 lf') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite H10; apply (H22 0%nat);
    [ simpl; apply lt_O_Sn
      | unfold open_interval; simpl; replace s1 with t2;
        [ elim H6; intros; split; assumption
          | simpl in H19; rewrite H19; rewrite Hyp_min; reflexivity ] ].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i < S (Rlength s3))%nat
x:R
H6:pos_Rl (cons t1 (cons s2 s3)) (S i) < x <
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i < x <
pos_Rl (cons t1 (cons s2 s3)) (S i) ->
f x = pos_Rl (cons r1 lf') i
f x = pos_Rl (cons r1 lf') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; simpl in H6; apply (H22 (S i));
    [ simpl; assumption
      | unfold open_interval; simpl; apply H6 ].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 lf')))%nat ->
pos_Rl (cons r1 lf') i <> pos_Rl (cons r1 lf') (S i) \/
f (pos_Rl (cons t1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r1 lf') i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  intros; simpl in H1; rewrite H10;
    change
      (pos_Rl (cons r2 lf') i <> pos_Rl (cons r2 lf') (S i) \/
        f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <> pos_Rl (cons r2 lf') i)
     ; rewrite <- H9; elim H8; intros; apply H6;
        simpl; apply H1.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  intros; induction  i as [| i Hreci].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 <
 Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat
pos_Rl (cons t1 (cons s2 s3)) 0 <>
pos_Rl (cons t1 (cons s2 s3)) 1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i <
 Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat
Hreci:(i <
 Init.Nat.pred
   (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
pos_Rl (cons t1 (cons s2 s3)) (S i) <>
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; red; intro; elim Hyp_eq; apply Rle_antisym.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 <
 Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat
H6:t1 = s2
t1 <= t2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 <
 Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat
H6:t1 = s2
t2 <= t1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i <
 Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat
Hreci:(i <
 Init.Nat.pred
   (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
pos_Rl (cons t1 (cons s2 s3)) (S i) <>
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  apply (H12 0%nat); simpl; apply lt_O_Sn.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H1:(0 <
 Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat
H6:t1 = s2
t2 <= t1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i <
 Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat
Hreci:(i <
 Init.Nat.pred
   (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
pos_Rl (cons t1 (cons s2 s3)) (S i) <>
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite <- Hyp_min; rewrite H6; simpl in H19; rewrite <- H19;
    apply (H16 0%nat); simpl; apply lt_O_Sn.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 = r1
H12:ordered_Rlist (cons t1 (cons t2 t3))
H14:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H13:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H15:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
H16:ordered_Rlist (cons s1 (cons s2 s3))
H19:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H18:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H20:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
i:nat
H1:(S i <
 Init.Nat.pred (Rlength (cons t1 (cons s2 s3))))%nat
Hreci:(i <
 Init.Nat.pred
   (Rlength (cons t1 (cons s2 s3))))%nat ->
pos_Rl (cons t1 (cons s2 s3)) i <>
pos_Rl (cons t1 (cons s2 s3)) (S i)
pos_Rl (cons t1 (cons s2 s3)) (S i) <>
pos_Rl (cons t1 (cons s2 s3)) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  elim H8; intros; rewrite H9 in H21; apply (H21 (S i)); simpl;
    simpl in H1; apply H1.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  exists (cons t1 l'); exists (cons r1 (cons r2 lf')); rewrite H9 in H6;
    rewrite H3 in H1; unfold adapted_couple in H1, H6;
      decompose [and] H6; decompose [and] H1; clear H6 H1;
        unfold adapted_couple_opt; unfold adapted_couple;
          repeat split.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
ordered_Rlist (cons t1 l')
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l') 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite H9; unfold ordered_Rlist; intros; simpl in H1;
    induction  i as [| i Hreci].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < S (S (Rlength s3)))%nat
pos_Rl (cons t1 (cons s1 (cons s2 s3))) 0 <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) 1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (S (Rlength s3)))%nat
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l') 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; replace s1 with t2.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < S (S (Rlength s3)))%nat
t1 <= t2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < S (S (Rlength s3)))%nat
t2 = s1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (S (Rlength s3)))%nat
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l') 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  apply (H16 0%nat); simpl; apply lt_O_Sn.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < S (S (Rlength s3)))%nat
t2 = s1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (S (Rlength s3)))%nat
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l') 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl in H14; rewrite H14; rewrite Hyp_min; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (S (Rlength s3)))%nat
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l') 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  change
    (pos_Rl (cons s1 (cons s2 s3)) i <= pos_Rl (cons s1 (cons s2 s3)) (S i))
   ; apply (H12 i); simpl; apply lt_S_n;
      assumption.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l') 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; simpl in H19; apply H19.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite H9; simpl; simpl in H13; rewrite H13; unfold Rmax;
    decide (Rle_dec t2 b) with H5; decide (Rle_dec a b) with H0; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite H9; simpl; simpl in H15; rewrite H15; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  intros; simpl in H1; unfold constant_D_eq, open_interval; intros;
    induction  i as [| i Hreci].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') 0 < x < pos_Rl (cons t1 l') 1
f x = pos_Rl (cons r1 (cons r2 lf')) 0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
Hreci:(i < Rlength l')%nat ->
pos_Rl (cons t1 l') i < x <
pos_Rl (cons t1 l') (S i) ->
f x = pos_Rl (cons r1 (cons r2 lf')) i
f x = pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; rewrite H9 in H6; simpl in H6; apply (H22 0%nat).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:t1 < x < s1
(0 < Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:t1 < x < s1
open_interval (pos_Rl (cons t1 (cons t2 t3)) 0)
  (pos_Rl (cons t1 (cons t2 t3)) 1) x
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
Hreci:(i < Rlength l')%nat ->
pos_Rl (cons t1 l') i < x <
pos_Rl (cons t1 l') (S i) ->
f x = pos_Rl (cons r1 (cons r2 lf')) i
f x = pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; apply lt_O_Sn.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:t1 < x < s1
open_interval (pos_Rl (cons t1 (cons t2 t3)) 0)
  (pos_Rl (cons t1 (cons t2 t3)) 1) x
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
Hreci:(i < Rlength l')%nat ->
pos_Rl (cons t1 l') i < x <
pos_Rl (cons t1 l') (S i) ->
f x = pos_Rl (cons r1 (cons r2 lf')) i
f x = pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  unfold open_interval; simpl.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:t1 < x < s1
t1 < x < t2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
Hreci:(i < Rlength l')%nat ->
pos_Rl (cons t1 l') i < x <
pos_Rl (cons t1 l') (S i) ->
f x = pos_Rl (cons r1 (cons r2 lf')) i
f x = pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  replace t2 with s1.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:t1 < x < s1
t1 < x < s1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:t1 < x < s1
s1 = t2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
Hreci:(i < Rlength l')%nat ->
pos_Rl (cons t1 l') i < x <
pos_Rl (cons t1 l') (S i) ->
f x = pos_Rl (cons r1 (cons r2 lf')) i
f x = pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  assumption.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:t1 < x < s1
s1 = t2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
Hreci:(i < Rlength l')%nat ->
pos_Rl (cons t1 l') i < x <
pos_Rl (cons t1 l') (S i) ->
f x = pos_Rl (cons r1 (cons r2 lf')) i
f x = pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl in H14; rewrite H14; rewrite Hyp_min; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
Hreci:(i < Rlength l')%nat ->
pos_Rl (cons t1 l') i < x <
pos_Rl (cons t1 l') (S i) ->
f x = pos_Rl (cons r1 (cons r2 lf')) i
f x = pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  change (f x = pos_Rl (cons r2 lf') i); clear Hreci; apply (H17 i).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
(i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
open_interval (pos_Rl (cons s1 (cons s2 s3)) i)
  (pos_Rl (cons s1 (cons s2 s3)) (S i)) x
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; rewrite H9 in H1; simpl in H1; apply lt_S_n; apply H1.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
open_interval (pos_Rl (cons s1 (cons s2 s3)) i)
  (pos_Rl (cons s1 (cons s2 s3)) (S i)) x
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite H9 in H6; unfold open_interval; apply H6.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  intros; simpl in H1; induction  i as [| i Hreci].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < S (Rlength lf'))%nat
pos_Rl (cons r1 (cons r2 lf')) 0 <>
pos_Rl (cons r1 (cons r2 lf')) 1 \/
f (pos_Rl (cons t1 l') 1) <>
pos_Rl (cons r1 (cons r2 lf')) 0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (Rlength lf'))%nat
Hreci:(i < S (Rlength lf'))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
pos_Rl (cons r1 (cons r2 lf')) (S i) <>
pos_Rl (cons r1 (cons r2 lf')) (S (S i)) \/
f (pos_Rl (cons t1 l') (S (S i))) <>
pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; rewrite H9; right; simpl; replace s1 with t2.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < S (Rlength lf'))%nat
f t2 <> r1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < S (Rlength lf'))%nat
t2 = s1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (Rlength lf'))%nat
Hreci:(i < S (Rlength lf'))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
pos_Rl (cons r1 (cons r2 lf')) (S i) <>
pos_Rl (cons r1 (cons r2 lf')) (S (S i)) \/
f (pos_Rl (cons t1 l') (S (S i))) <>
pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  assumption.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < S (Rlength lf'))%nat
t2 = s1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (Rlength lf'))%nat
Hreci:(i < S (Rlength lf'))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
pos_Rl (cons r1 (cons r2 lf')) (S i) <>
pos_Rl (cons r1 (cons r2 lf')) (S (S i)) \/
f (pos_Rl (cons t1 l') (S (S i))) <>
pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl in H14; rewrite H14; rewrite Hyp_min; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (Rlength lf'))%nat
Hreci:(i < S (Rlength lf'))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
pos_Rl (cons r1 (cons r2 lf')) (S i) <>
pos_Rl (cons r1 (cons r2 lf')) (S (S i)) \/
f (pos_Rl (cons t1 l') (S (S i))) <>
pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  elim H8; intros; apply (H6 i).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (Rlength lf'))%nat
Hreci:(i < S (Rlength lf'))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
pos_Rl (cons r2 lf') i0 <>
pos_Rl (cons r2 lf') (S i0) \/
f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0
H21:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l'))%nat ->
pos_Rl l' i0 <> pos_Rl l' (S i0)
(i < Init.Nat.pred (Rlength (cons r2 lf')))%nat
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; apply lt_S_n; apply H1.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  intros; rewrite H9; induction  i as [| i Hreci].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Init.Nat.pred (Rlength (cons t1 l')))%nat
pos_Rl (cons t1 (cons s1 (cons s2 s3))) 0 <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) 1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons t1 l')))%nat
Hreci:(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; red; intro; elim Hyp_eq; apply Rle_antisym.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Init.Nat.pred (Rlength (cons t1 l')))%nat
H6:t1 = s1
t1 <= t2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Init.Nat.pred (Rlength (cons t1 l')))%nat
H6:t1 = s1
t2 <= t1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons t1 l')))%nat
Hreci:(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  apply (H16 0%nat); simpl; apply lt_O_Sn.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Init.Nat.pred (Rlength (cons t1 l')))%nat
H6:t1 = s1
t2 <= t1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons t1 l')))%nat
Hreci:(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite <- Hyp_min; rewrite H6; simpl in H14; rewrite <- H14; right;
    reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 = r2
H11:f t2 <> r1
H12:ordered_Rlist (cons s1 (cons s2 s3))
H14:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H13:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H15:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H17:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H16:ordered_Rlist (cons t1 (cons t2 t3))
H19:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H18:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H20:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H22:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons t1 l')))%nat
Hreci:(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  elim H8; intros; rewrite <- H9; apply (H21 i); rewrite H9; rewrite H9 in H1;
    simpl; simpl in H1; apply lt_S_n; apply H1.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
H6:adapted_couple f t2 b l' (cons r2 lf')
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
s1, s2:R
s3:Rlist
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a b l'0 lf'0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  exists (cons t1 l'); exists (cons r1 (cons r2 lf')); rewrite H9 in H6;
    rewrite H3 in H1; unfold adapted_couple in H1, H6;
      decompose [and] H6; decompose [and] H1; clear H6 H1;
        unfold adapted_couple_opt; unfold adapted_couple;
          repeat split.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
ordered_Rlist (cons t1 l')
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l') 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite H9; unfold ordered_Rlist; intros; simpl in H1;
    induction  i as [| i Hreci].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < S (S (Rlength s3)))%nat
pos_Rl (cons t1 (cons s1 (cons s2 s3))) 0 <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) 1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (S (Rlength s3)))%nat
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l') 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; replace s1 with t2.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < S (S (Rlength s3)))%nat
t1 <= t2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < S (S (Rlength s3)))%nat
t2 = s1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (S (Rlength s3)))%nat
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l') 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  apply (H15 0%nat); simpl; apply lt_O_Sn.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < S (S (Rlength s3)))%nat
t2 = s1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (S (Rlength s3)))%nat
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l') 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl in H13; rewrite H13; rewrite Hyp_min; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (S (Rlength s3)))%nat
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <=
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l') 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  change
    (pos_Rl (cons s1 (cons s2 s3)) i <= pos_Rl (cons s1 (cons s2 s3)) (S i))
   ; apply (H11 i); simpl; apply lt_S_n;
      assumption.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l') 0 = Rmin a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; simpl in H18; apply H18.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
pos_Rl (cons t1 l')
  (Init.Nat.pred (Rlength (cons t1 l'))) = 
Rmax a b
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite H9; simpl; simpl in H12; rewrite H12; unfold Rmax;
    decide (Rle_dec t2 b) with H5; decide (Rle_dec a b) with H0; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
Rlength (cons t1 l') =
S (Rlength (cons r1 (cons r2 lf')))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite H9; simpl; simpl in H14; rewrite H14; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons t1 l') i)
     (pos_Rl (cons t1 l') (S i)))
  (pos_Rl (cons r1 (cons r2 lf')) i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  intros; simpl in H1; unfold constant_D_eq, open_interval; intros;
    induction  i as [| i Hreci].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') 0 < x < pos_Rl (cons t1 l') 1
f x = pos_Rl (cons r1 (cons r2 lf')) 0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
Hreci:(i < Rlength l')%nat ->
pos_Rl (cons t1 l') i < x <
pos_Rl (cons t1 l') (S i) ->
f x = pos_Rl (cons r1 (cons r2 lf')) i
f x = pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; rewrite H9 in H6; simpl in H6; apply (H21 0%nat).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:t1 < x < s1
(0 < Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:t1 < x < s1
open_interval (pos_Rl (cons t1 (cons t2 t3)) 0)
  (pos_Rl (cons t1 (cons t2 t3)) 1) x
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
Hreci:(i < Rlength l')%nat ->
pos_Rl (cons t1 l') i < x <
pos_Rl (cons t1 l') (S i) ->
f x = pos_Rl (cons r1 (cons r2 lf')) i
f x = pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; apply lt_O_Sn.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:t1 < x < s1
open_interval (pos_Rl (cons t1 (cons t2 t3)) 0)
  (pos_Rl (cons t1 (cons t2 t3)) 1) x
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
Hreci:(i < Rlength l')%nat ->
pos_Rl (cons t1 l') i < x <
pos_Rl (cons t1 l') (S i) ->
f x = pos_Rl (cons r1 (cons r2 lf')) i
f x = pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  unfold open_interval; simpl; replace t2 with s1.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:t1 < x < s1
t1 < x < s1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:t1 < x < s1
s1 = t2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
Hreci:(i < Rlength l')%nat ->
pos_Rl (cons t1 l') i < x <
pos_Rl (cons t1 l') (S i) ->
f x = pos_Rl (cons r1 (cons r2 lf')) i
f x = pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  assumption.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Rlength l')%nat
x:R
H6:t1 < x < s1
s1 = t2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
Hreci:(i < Rlength l')%nat ->
pos_Rl (cons t1 l') i < x <
pos_Rl (cons t1 l') (S i) ->
f x = pos_Rl (cons r1 (cons r2 lf')) i
f x = pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl in H13; rewrite H13; rewrite Hyp_min; reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
Hreci:(i < Rlength l')%nat ->
pos_Rl (cons t1 l') i < x <
pos_Rl (cons t1 l') (S i) ->
f x = pos_Rl (cons r1 (cons r2 lf')) i
f x = pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  change (f x = pos_Rl (cons r2 lf') i); clear Hreci; apply (H16 i).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
(i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
open_interval (pos_Rl (cons s1 (cons s2 s3)) i)
  (pos_Rl (cons s1 (cons s2 s3)) (S i)) x
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; rewrite H9 in H1; simpl in H1; apply lt_S_n; apply H1.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Rlength l')%nat
x:R
H6:pos_Rl (cons t1 l') (S i) < x <
pos_Rl (cons t1 l') (S (S i))
open_interval (pos_Rl (cons s1 (cons s2 s3)) i)
  (pos_Rl (cons s1 (cons s2 s3)) (S i)) x
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite H9 in H6; unfold open_interval; apply H6.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons r2 lf'))))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  intros; simpl in H1; induction  i as [| i Hreci].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < S (Rlength lf'))%nat
pos_Rl (cons r1 (cons r2 lf')) 0 <>
pos_Rl (cons r1 (cons r2 lf')) 1 \/
f (pos_Rl (cons t1 l') 1) <>
pos_Rl (cons r1 (cons r2 lf')) 0
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (Rlength lf'))%nat
Hreci:(i < S (Rlength lf'))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
pos_Rl (cons r1 (cons r2 lf')) (S i) <>
pos_Rl (cons r1 (cons r2 lf')) (S (S i)) \/
f (pos_Rl (cons t1 l') (S (S i))) <>
pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; left; assumption.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (Rlength lf'))%nat
Hreci:(i < S (Rlength lf'))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
pos_Rl (cons r1 (cons r2 lf')) (S i) <>
pos_Rl (cons r1 (cons r2 lf')) (S (S i)) \/
f (pos_Rl (cons t1 l') (S (S i))) <>
pos_Rl (cons r1 (cons r2 lf')) (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  elim H8; intros; apply (H6 i).f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < S (Rlength lf'))%nat
Hreci:(i < S (Rlength lf'))%nat ->
pos_Rl (cons r1 (cons r2 lf')) i <>
pos_Rl (cons r1 (cons r2 lf')) (S i) \/
f (pos_Rl (cons t1 l') (S i)) <>
pos_Rl (cons r1 (cons r2 lf')) i
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
pos_Rl (cons r2 lf') i0 <>
pos_Rl (cons r2 lf') (S i0) \/
f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0
H20:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l'))%nat ->
pos_Rl l' i0 <> pos_Rl l' (S i0)
(i < Init.Nat.pred (Rlength (cons r2 lf')))%nat
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; apply lt_S_n; apply H1.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 l') i <> pos_Rl (cons t1 l') (S i)
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  intros; rewrite H9; induction  i as [| i Hreci].f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Init.Nat.pred (Rlength (cons t1 l')))%nat
pos_Rl (cons t1 (cons s1 (cons s2 s3))) 0 <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) 1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons t1 l')))%nat
Hreci:(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  simpl; red; intro; elim Hyp_eq; apply Rle_antisym.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Init.Nat.pred (Rlength (cons t1 l')))%nat
H6:t1 = s1
t1 <= t2
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Init.Nat.pred (Rlength (cons t1 l')))%nat
H6:t1 = s1
t2 <= t1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons t1 l')))%nat
Hreci:(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  apply (H15 0%nat); simpl; apply lt_O_Sn.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i <>
 pos_Rl (cons r2 lf') (S i) \/
 f (pos_Rl l' (S i)) <> pos_Rl (cons r2 lf') i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i <> pos_Rl l' (S i))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r2 lf') i)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i)
     (pos_Rl (cons t1 (cons t2 t3)) (S i)))
  (pos_Rl (cons r1 lf) i)
H1:(0 < Init.Nat.pred (Rlength (cons t1 l')))%nat
H6:t1 = s1
t2 <= t1
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons t1 l')))%nat
Hreci:(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite <- Hyp_min; rewrite H6; simpl in H13; rewrite <- H13; right;
    reflexivity.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l'0 lf'0 : Rlist,
  adapted_couple_opt f a0 b0 l'0 lf'0
r1:R
lf:Rlist
a, b:R
H0:a <= b
t1, t2:R
t3:Rlist
H2:a <> b
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
H5:t2 <= b
Hyp_eq:t1 <> t2
l':Rlist
r2:R
lf':Rlist
H7:t2 <> b
Hyp_min:Rmin t2 b = t2
s1, s2:R
s3:Rlist
H8:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r2 lf')))%nat ->
 pos_Rl (cons r2 lf') i0 <>
 pos_Rl (cons r2 lf') (S i0) \/
 f (pos_Rl l' (S i0)) <> pos_Rl (cons r2 lf') i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l'))%nat ->
 pos_Rl l' i0 <> pos_Rl l' (S i0))
H9:l' = cons s1 (cons s2 s3)
H10:r1 <> r2
H11:ordered_Rlist (cons s1 (cons s2 s3))
H13:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin t2 b
H12:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax t2 b
H14:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r2 lf'))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r2 lf') i0)
H15:ordered_Rlist (cons t1 (cons t2 t3))
H18:pos_Rl (cons t1 (cons t2 t3)) 0 = Rmin a b
H17:pos_Rl (cons t1 (cons t2 t3))
  (Init.Nat.pred
     (Rlength (cons t1 (cons t2 t3)))) =
Rmax a b
H19:Rlength (cons t1 (cons t2 t3)) =
S (Rlength (cons r1 lf))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons t1 (cons t2 t3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons t1 (cons t2 t3)) i0)
     (pos_Rl (cons t1 (cons t2 t3)) (S i0)))
  (pos_Rl (cons r1 lf) i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons t1 l')))%nat
Hreci:(i < Init.Nat.pred (Rlength (cons t1 l')))%nat ->
pos_Rl (cons t1 (cons s1 (cons s2 s3))) i <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i)
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S i) <>
pos_Rl (cons t1 (cons s1 (cons s2 s3))) (S (S i))
f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  elim H8; intros; rewrite <- H9; apply (H20 i); rewrite H9; rewrite H9 in H1;
    simpl; simpl in H1; apply lt_S_n; apply H1.f:R -> R
l:Rlist
r:R
r0:Rlist
H:forall (lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 r0 lf0 ->
exists l' lf' : Rlist,
  adapted_couple_opt f a0 b0 l' lf'
r1:R
lf:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r r0) (cons r1 lf)
H2:a <> b
t1, t2:R
t3:Rlist
H3:cons r r0 = cons t1 (cons t2 t3)
H4:adapted_couple f t2 b r0 lf
t2 <= b
  rewrite H3 in H1; clear H4; unfold adapted_couple in H1; decompose [and] H1;
    clear H1; clear H H7 H9; cut (Rmax a b = b);
      [ intro; rewrite H in H5; rewrite <- H5; apply RList_P7;
        [ assumption | simpl; right; left; reflexivity ]
        | unfold Rmax; decide (Rle_dec a b) with H0; reflexivity ].
Qed.

Lemma StepFun_P11 :
  forall (a b r r1 r3 s1 s2 r4:R) (r2 lf1 s3 lf2:Rlist)
    (f:R -> R),
    a < b ->
    adapted_couple f a b (cons r (cons r1 r2)) (cons r3 lf1) ->
    adapted_couple_opt f a b (cons s1 (cons s2 s3)) (cons r4 lf2) -> r1 <= s2.forall (a b r r1 r3 s1 s2 r4 : R)
  (r2 lf1 s3 lf2 : Rlist) (f : R -> R),
a < b ->
adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1) ->
adapted_couple_opt f a b (cons s1 (cons s2 s3))
  (cons r4 lf2) -> r1 <= s2
Proof.forall (a b r r1 r3 s1 s2 r4 : R)
  (r2 lf1 s3 lf2 : Rlist) (f : R -> R),
a < b ->
adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1) ->
adapted_couple_opt f a b (cons s1 (cons s2 s3))
  (cons r4 lf2) -> r1 <= s2
  intros; unfold adapted_couple_opt in H1; elim H1; clear H1; intros;
    unfold adapted_couple in H0, H1; decompose [and] H0;
      decompose [and] H1; clear H0 H1; assert (H12 : r = s1).a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) = S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
(open_interval (pos_Rl (cons s1 (cons s2 s3)) i)
(pos_Rl (cons s1 (cons s2 s3)) (S i))) (pos_Rl (cons r4 lf2) i)
r = s1
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) = S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
(open_interval (pos_Rl (cons s1 (cons s2 s3)) i)
(pos_Rl (cons s1 (cons s2 s3)) (S i))) (pos_Rl (cons r4 lf2) i)
H12:r = s1
r1 <= s2
  simpl in H10; simpl in H5; rewrite H10; rewrite H5; reflexivity.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) = S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
(open_interval (pos_Rl (cons s1 (cons s2 s3)) i)
(pos_Rl (cons s1 (cons s2 s3)) (S i))) (pos_Rl (cons r4 lf2) i)
H12:r = s1
r1 <= s2
  assert (H14 := H3 0%nat (lt_O_Sn _)); simpl in H14; elim H14; intro.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) = S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
(open_interval (pos_Rl (cons s1 (cons s2 s3)) i)
(pos_Rl (cons s1 (cons s2 s3)) (S i))) (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) = S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
(open_interval (pos_Rl (cons s1 (cons s2 s3)) i)
(pos_Rl (cons s1 (cons s2 s3)) (S i))) (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  assert (H15 := H7 0%nat (lt_O_Sn _)); simpl in H15; elim H15; intro.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) = S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
(open_interval (pos_Rl (cons s1 (cons s2 s3)) i)
(pos_Rl (cons s1 (cons s2 s3)) (S i))) (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 < s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) = S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
(open_interval (pos_Rl (cons s1 (cons s2 s3)) i)
(pos_Rl (cons s1 (cons s2 s3)) (S i))) (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) = S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
(open_interval (pos_Rl (cons s1 (cons s2 s3)) i)
(pos_Rl (cons s1 (cons s2 s3)) (S i))) (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  rewrite <- H12 in H1; destruct (Rle_dec r1 s2) as [Hle|Hnle]; try assumption.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) = S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
(open_interval (pos_Rl (cons s1 (cons s2 s3)) i)
(pos_Rl (cons s1 (cons s2 s3)) (S i))) (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  assert (H16 : s2 < r1); auto with real.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  induction  s3 as [| r0 s3 Hrecs3].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 nil)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 nil))))%nat ->
 pos_Rl (cons s1 (cons s2 nil)) i <>
 pos_Rl (cons s1 (cons s2 nil)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 nil))
H10:pos_Rl (cons s1 (cons s2 nil)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 nil))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 nil)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 nil)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 nil))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 nil)) i)
     (pos_Rl (cons s1 (cons s2 nil)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i))) (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
Hrecs3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i)) ->
ordered_Rlist (cons s1 (cons s2 s3)) ->
pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b ->
pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b ->
Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2)) ->
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 s3))))%nat ->
 constant_D_eq f
   (open_interval
      (pos_Rl (cons s1 (cons s2 s3)) i)
      (pos_Rl (cons s1 (cons s2 s3)) (S i)))
   (pos_Rl (cons r4 lf2) i)) -> 
r1 <= s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  simpl in H9; rewrite H9 in H16; cut (r1 <= Rmax a b).a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 nil)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 nil))))%nat ->
 pos_Rl (cons s1 (cons s2 nil)) i <>
 pos_Rl (cons s1 (cons s2 nil)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 nil))
H10:pos_Rl (cons s1 (cons s2 nil)) 0 = Rmin a b
H9:s2 = Rmax a b
H11:Rlength (cons s1 (cons s2 nil)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 nil))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 nil)) i)
     (pos_Rl (cons s1 (cons s2 nil)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:Rmax a b < r1
r1 <= Rmax a b -> r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 nil)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 nil))))%nat ->
 pos_Rl (cons s1 (cons s2 nil)) i <>
 pos_Rl (cons s1 (cons s2 nil)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 nil))
H10:pos_Rl (cons s1 (cons s2 nil)) 0 = Rmin a b
H9:s2 = Rmax a b
H11:Rlength (cons s1 (cons s2 nil)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 nil))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 nil)) i)
     (pos_Rl (cons s1 (cons s2 nil)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:Rmax a b < r1
r1 <= Rmax a b
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i))) (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
Hrecs3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i)) ->
ordered_Rlist (cons s1 (cons s2 s3)) ->
pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b ->
pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b ->
Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2)) ->
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 s3))))%nat ->
 constant_D_eq f
   (open_interval
      (pos_Rl (cons s1 (cons s2 s3)) i)
      (pos_Rl (cons s1 (cons s2 s3)) (S i)))
   (pos_Rl (cons r4 lf2) i)) -> 
r1 <= s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H17 H16)).a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 nil)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 nil))))%nat ->
 pos_Rl (cons s1 (cons s2 nil)) i <>
 pos_Rl (cons s1 (cons s2 nil)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 nil))
H10:pos_Rl (cons s1 (cons s2 nil)) 0 = Rmin a b
H9:s2 = Rmax a b
H11:Rlength (cons s1 (cons s2 nil)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 nil))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 nil)) i)
     (pos_Rl (cons s1 (cons s2 nil)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:Rmax a b < r1
r1 <= Rmax a b
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i))) (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
Hrecs3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i)) ->
ordered_Rlist (cons s1 (cons s2 s3)) ->
pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b ->
pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b ->
Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2)) ->
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 s3))))%nat ->
 constant_D_eq f
   (open_interval
      (pos_Rl (cons s1 (cons s2 s3)) i)
      (pos_Rl (cons s1 (cons s2 s3)) (S i)))
   (pos_Rl (cons r4 lf2) i)) -> 
r1 <= s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  rewrite <- H4; apply RList_P7;
    [ assumption | simpl; right; left; reflexivity ].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i))) (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
Hrecs3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i)) ->
ordered_Rlist (cons s1 (cons s2 s3)) ->
pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b ->
pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b ->
Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2)) ->
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 s3))))%nat ->
 constant_D_eq f
   (open_interval
      (pos_Rl (cons s1 (cons s2 s3)) i)
      (pos_Rl (cons s1 (cons s2 s3)) (S i)))
   (pos_Rl (cons r4 lf2) i)) -> 
r1 <= s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  clear Hrecs3; induction  lf2 as [| r5 lf2 Hreclf2].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 nil)))%nat ->
 pos_Rl (cons r4 nil) i <>
 pos_Rl (cons r4 nil) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 nil) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 nil))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i))) (pos_Rl (cons r4 nil) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
Hreclf2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
      (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength
       (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3)))
   (S i)) ->
Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 lf2)) ->
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength
       (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 constant_D_eq f
   (open_interval
      (pos_Rl
         (cons s1 (cons s2 (cons r0 s3))) i)
      (pos_Rl
         (cons s1 (cons s2 (cons r0 s3)))
         (S i))) (pos_Rl (cons r4 lf2) i)) ->
r1 <= s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  simpl in H11; discriminate.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
Hreclf2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
      (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength
       (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3)))
   (S i)) ->
Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 lf2)) ->
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength
       (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 constant_D_eq f
   (open_interval
      (pos_Rl
         (cons s1 (cons s2 (cons r0 s3))) i)
      (pos_Rl
         (cons s1 (cons s2 (cons r0 s3)))
         (S i))) (pos_Rl (cons r4 lf2) i)) ->
r1 <= s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  clear Hreclf2; assert (H17 : r3 = r4).a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
r3 = r4
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  set (x := (r + s2) / 2); assert (H17 := H8 0%nat (lt_O_Sn _));
    assert (H18 := H13 0%nat (lt_O_Sn _));
      unfold constant_D_eq, open_interval in H17, H18; simpl in H17;
        simpl in H18; rewrite <- (H17 x).a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
f x = r4
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
r < x < r1
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  rewrite <- (H18 x).a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
f x = f x
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
s1 < x < s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
r < x < r1
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  reflexivity.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
s1 < x < s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
r < x < r1
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  rewrite <- H12; unfold x; split.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
r < (r + s2) / 2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
(r + s2) / 2 < s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
r < x < r1
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
            | discrR ] ].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
(r + s2) / 2 < s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
r < x < r1
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite (Rplus_comm r); rewrite double;
            apply Rplus_lt_compat_l; assumption
            | discrR ] ].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
r < x < r1
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  unfold x; split.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
r < (r + s2) / 2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
(r + s2) / 2 < r1
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
            | discrR ] ].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
x:=(r + s2) / 2:R
H17:forall x0 : R, r < x0 < r1 -> f x0 = r3
H18:forall x0 : R, s1 < x0 < s2 -> f x0 = r4
(r + s2) / 2 < r1
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  apply Rlt_trans with s2;
    [ apply Rmult_lt_reg_l with 2;
      [ prove_sup0
        | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
          rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
            [ rewrite Rmult_1_l; rewrite (Rplus_comm r); rewrite double;
              apply Rplus_lt_compat_l; assumption
              | discrR ] ]
      | assumption ].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  assert (H18 : f s2 = r3).a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
f s2 = r3
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  apply (H8 0%nat);
    [ simpl; apply lt_O_Sn
      | unfold open_interval; simpl; split; assumption ].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  assert (H19 : r3 = r5).a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
r3 = r5
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:r3 = r5
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  assert (H19 := H7 1%nat); simpl in H19;
    assert (H20 := H19 (lt_n_S _ _ (lt_O_Sn _))); elim H20;
      intro.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
r3 = r5
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 = r0
r3 = r5
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:r3 = r5
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  set (x := (s2 + Rmin r1 r0) / 2); assert (H22 := H8 0%nat);
    assert (H23 := H13 1%nat); simpl in H22; simpl in H23;
      rewrite <- (H22 (lt_O_Sn _) x).a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
f x = r5
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
open_interval r r1 x
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 = r0
r3 = r5
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:r3 = r5
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  rewrite <- (H23 (lt_n_S _ _ (lt_O_Sn _)) x).a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
f x = f x
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
open_interval s2 r0 x
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
open_interval r r1 x
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 = r0
r3 = r5
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:r3 = r5
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  reflexivity.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
open_interval s2 r0 x
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
open_interval r r1 x
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 = r0
r3 = r5
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:r3 = r5
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  unfold open_interval; simpl; unfold x; split.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
s2 < (s2 + Rmin r1 r0) / 2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
(s2 + Rmin r1 r0) / 2 < r0
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
open_interval r r1 x
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 = r0
r3 = r5
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:r3 = r5
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l;
            unfold Rmin; case (Rle_dec r1 r0); intro;
              assumption
            | discrR ] ].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
(s2 + Rmin r1 r0) / 2 < r0
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
open_interval r r1 x
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 = r0
r3 = r5
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:r3 = r5
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double;
            apply Rlt_le_trans with (r0 + Rmin r1 r0);
              [ do 2 rewrite <- (Rplus_comm (Rmin r1 r0)); apply Rplus_lt_compat_l;
                assumption
                | apply Rplus_le_compat_l; apply Rmin_r ]
            | discrR ] ].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
open_interval r r1 x
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 = r0
r3 = r5
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:r3 = r5
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  unfold open_interval; simpl; unfold x; split.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
r < (s2 + Rmin r1 r0) / 2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
(s2 + Rmin r1 r0) / 2 < r1
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 = r0
r3 = r5
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:r3 = r5
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  apply Rlt_trans with s2;
    [ assumption
      | apply Rmult_lt_reg_l with 2;
        [ prove_sup0
          | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
            rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
              [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l;
                unfold Rmin; case (Rle_dec r1 r0);
                  intro; assumption
                | discrR ] ] ].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 < r0
x:=(s2 + Rmin r1 r0) / 2:R
H22:(0 < S (Rlength r2))%nat ->
constant_D_eq f (open_interval r r1) r3
H23:(1 < S (S (Rlength s3)))%nat ->
constant_D_eq f (open_interval s2 r0) r5
(s2 + Rmin r1 r0) / 2 < r1
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 = r0
r3 = r5
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:r3 = r5
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double;
            apply Rlt_le_trans with (r1 + Rmin r1 r0);
              [ do 2 rewrite <- (Rplus_comm (Rmin r1 r0)); apply Rplus_lt_compat_l;
                assumption
                | apply Rplus_le_compat_l; apply Rmin_l ]
            | discrR ] ].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:(1 < S (S (Rlength s3)))%nat -> s2 <= r0
H20:s2 <= r0
H21:s2 = r0
r3 = r5
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:r3 = r5
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  elim H2; clear H2; intros; assert (H23 := H22 1%nat); simpl in H23;
    assert (H24 := H23 (lt_n_S _ _ (lt_O_Sn _))); elim H24;
      assumption.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1:Rlist
r0:R
s3:Rlist
r5:R
lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons r4 (cons r5 lf2))))%nat ->
 pos_Rl (cons r4 (cons r5 lf2)) i <>
 pos_Rl (cons r4 (cons r5 lf2)) (S i) \/
 f
   (pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i)) <>
 pos_Rl (cons r4 (cons r5 lf2)) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred
    (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) i <>
 pos_Rl (cons s1 (cons s2 (cons r0 s3))) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 (cons r0 s3)))
H10:pos_Rl (cons s1 (cons s2 (cons r0 s3))) 0 =
Rmin a b
H9:pos_Rl (cons s1 (cons s2 (cons r0 s3)))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 (cons r0 s3))))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 (cons r0 s3))) =
S (Rlength (cons r4 (cons r5 lf2)))
H13:forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons s1 (cons s2 (cons r0 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 (cons r0 s3))) i)
     (pos_Rl (cons s1 (cons s2 (cons r0 s3)))
        (S i)))
  (pos_Rl (cons r4 (cons r5 lf2)) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:r < s2
Hnle:~ r1 <= s2
H16:s2 < r1
H17:r3 = r4
H18:f s2 = r3
H19:r3 = r5
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  elim H2; intros; assert (H22 := H20 0%nat); simpl in H22;
    assert (H23 := H22 (lt_O_Sn _)); elim H23; intro;
      [ elim H24; rewrite <- H17; rewrite <- H19; reflexivity
        | elim H24; rewrite <- H17; assumption ].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r < r1
H15:s1 <= s2
H1:s1 = s2
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  elim H2; clear H2; intros; assert (H17 := H16 0%nat); simpl in H17;
    elim (H17 (lt_O_Sn _)); assumption.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a < b
H2:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i <
  Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
H3:ordered_Rlist (cons r (cons r1 r2))
H5:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H6:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H8:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H7:ordered_Rlist (cons s1 (cons s2 s3))
H10:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H9:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H11:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H13:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H12:r = s1
H14:r <= r1
H0:r = r1
r1 <= s2
  rewrite <- H0; rewrite H12; apply (H7 0%nat); simpl; apply lt_O_Sn.
Qed.

Lemma StepFun_P12 :
  forall (a b:R) (f:R -> R) (l lf:Rlist),
    adapted_couple_opt f a b l lf -> adapted_couple_opt f b a l lf.forall (a b : R) (f : R -> R) (l lf : Rlist),
adapted_couple_opt f a b l lf ->
adapted_couple_opt f b a l lf
Proof.forall (a b : R) (f : R -> R) (l lf : Rlist),
adapted_couple_opt f a b l lf ->
adapted_couple_opt f b a l lf
  unfold adapted_couple_opt; unfold adapted_couple; intros;
    decompose [and] H; clear H; repeat split; try assumption.a, b:R
f:R -> R
l, lf:Rlist
H2:ordered_Rlist l
H0:pos_Rl l 0 = Rmin a b
H3:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H4:Rlength l = S (Rlength lf)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
pos_Rl lf i <> pos_Rl lf (S i) \/
f (pos_Rl l (S i)) <> pos_Rl lf i
H7:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
pos_Rl l i <> pos_Rl l (S i)
pos_Rl l 0 = Rmin b a
a, b:R
f:R -> R
l, lf:Rlist
H2:ordered_Rlist l
H0:pos_Rl l 0 = Rmin a b
H3:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H4:Rlength l = S (Rlength lf)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
pos_Rl lf i <> pos_Rl lf (S i) \/
f (pos_Rl l (S i)) <> pos_Rl lf i
H7:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
pos_Rl l i <> pos_Rl l (S i)
pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax b a
  rewrite H0; unfold Rmin; case (Rle_dec a b); intro;
    case (Rle_dec b a); intro; try reflexivity.a, b:R
f:R -> R
l, lf:Rlist
H2:ordered_Rlist l
H0:pos_Rl l 0 = Rmin a b
H3:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H4:Rlength l = S (Rlength lf)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
pos_Rl lf i <> pos_Rl lf (S i) \/
f (pos_Rl l (S i)) <> pos_Rl lf i
H7:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
pos_Rl l i <> pos_Rl l (S i)
r:a <= b
r0:b <= a
a = b
a, b:R
f:R -> R
l, lf:Rlist
H2:ordered_Rlist l
H0:pos_Rl l 0 = Rmin a b
H3:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H4:Rlength l = S (Rlength lf)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
pos_Rl lf i <> pos_Rl lf (S i) \/
f (pos_Rl l (S i)) <> pos_Rl lf i
H7:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
pos_Rl l i <> pos_Rl l (S i)
n:~ a <= b
n0:~ b <= a
b = a
a, b:R
f:R -> R
l, lf:Rlist
H2:ordered_Rlist l
H0:pos_Rl l 0 = Rmin a b
H3:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H4:Rlength l = S (Rlength lf)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
pos_Rl lf i <> pos_Rl lf (S i) \/
f (pos_Rl l (S i)) <> pos_Rl lf i
H7:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
pos_Rl l i <> pos_Rl l (S i)
pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax b a
  apply Rle_antisym; assumption.a, b:R
f:R -> R
l, lf:Rlist
H2:ordered_Rlist l
H0:pos_Rl l 0 = Rmin a b
H3:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H4:Rlength l = S (Rlength lf)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
pos_Rl lf i <> pos_Rl lf (S i) \/
f (pos_Rl l (S i)) <> pos_Rl lf i
H7:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
pos_Rl l i <> pos_Rl l (S i)
n:~ a <= b
n0:~ b <= a
b = a
a, b:R
f:R -> R
l, lf:Rlist
H2:ordered_Rlist l
H0:pos_Rl l 0 = Rmin a b
H3:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H4:Rlength l = S (Rlength lf)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
pos_Rl lf i <> pos_Rl lf (S i) \/
f (pos_Rl l (S i)) <> pos_Rl lf i
H7:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
pos_Rl l i <> pos_Rl l (S i)
pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax b a
  apply Rle_antisym; auto with real.a, b:R
f:R -> R
l, lf:Rlist
H2:ordered_Rlist l
H0:pos_Rl l 0 = Rmin a b
H3:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H4:Rlength l = S (Rlength lf)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
pos_Rl lf i <> pos_Rl lf (S i) \/
f (pos_Rl l (S i)) <> pos_Rl lf i
H7:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
pos_Rl l i <> pos_Rl l (S i)
pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax b a
  rewrite H3; unfold Rmax; case (Rle_dec a b); intro;
    case (Rle_dec b a); intro; try reflexivity.a, b:R
f:R -> R
l, lf:Rlist
H2:ordered_Rlist l
H0:pos_Rl l 0 = Rmin a b
H3:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H4:Rlength l = S (Rlength lf)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
pos_Rl lf i <> pos_Rl lf (S i) \/
f (pos_Rl l (S i)) <> pos_Rl lf i
H7:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
pos_Rl l i <> pos_Rl l (S i)
r:a <= b
r0:b <= a
b = a
a, b:R
f:R -> R
l, lf:Rlist
H2:ordered_Rlist l
H0:pos_Rl l 0 = Rmin a b
H3:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H4:Rlength l = S (Rlength lf)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
pos_Rl lf i <> pos_Rl lf (S i) \/
f (pos_Rl l (S i)) <> pos_Rl lf i
H7:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
pos_Rl l i <> pos_Rl l (S i)
n:~ a <= b
n0:~ b <= a
a = b
  apply Rle_antisym; assumption.a, b:R
f:R -> R
l, lf:Rlist
H2:ordered_Rlist l
H0:pos_Rl l 0 = Rmin a b
H3:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H4:Rlength l = S (Rlength lf)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
pos_Rl lf i <> pos_Rl lf (S i) \/
f (pos_Rl l (S i)) <> pos_Rl lf i
H7:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
pos_Rl l i <> pos_Rl l (S i)
n:~ a <= b
n0:~ b <= a
a = b
  apply Rle_antisym; auto with real.
Qed.

Lemma StepFun_P13 :
  forall (a b r r1 r3 s1 s2 r4:R) (r2 lf1 s3 lf2:Rlist)
    (f:R -> R),
    a <> b ->
    adapted_couple f a b (cons r (cons r1 r2)) (cons r3 lf1) ->
    adapted_couple_opt f a b (cons s1 (cons s2 s3)) (cons r4 lf2) -> r1 <= s2.forall (a b r r1 r3 s1 s2 r4 : R)
  (r2 lf1 s3 lf2 : Rlist) (f : R -> R),
a <> b ->
adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1) ->
adapted_couple_opt f a b (cons s1 (cons s2 s3))
  (cons r4 lf2) -> r1 <= s2
Proof.forall (a b r r1 r3 s1 s2 r4 : R)
  (r2 lf1 s3 lf2 : Rlist) (f : R -> R),
a <> b ->
adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1) ->
adapted_couple_opt f a b (cons s1 (cons s2 s3))
  (cons r4 lf2) -> r1 <= s2
  intros; destruct (total_order_T a b) as [[Hlt|Heq]|Hgt].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a <> b
H0:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H1:adapted_couple_opt f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
Hlt:a < b
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a <> b
H0:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H1:adapted_couple_opt f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
Heq:a = b
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a <> b
H0:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H1:adapted_couple_opt f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
Hgt:a > b
r1 <= s2
  eapply StepFun_P11; [ apply Hlt | apply H0 | apply H1 ].a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a <> b
H0:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H1:adapted_couple_opt f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
Heq:a = b
r1 <= s2
a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a <> b
H0:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H1:adapted_couple_opt f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
Hgt:a > b
r1 <= s2
  elim H; assumption.a, b, r, r1, r3, s1, s2, r4:R
r2, lf1, s3, lf2:Rlist
f:R -> R
H:a <> b
H0:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H1:adapted_couple_opt f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
Hgt:a > b
r1 <= s2
  eapply StepFun_P11;
    [ apply Hgt | apply StepFun_P2; apply H0 | apply StepFun_P12; apply H1 ].
Qed.

Lemma StepFun_P14 :
  forall (f:R -> R) (l1 l2 lf1 lf2:Rlist) (a b:R),
    a <= b ->
    adapted_couple f a b l1 lf1 ->
    adapted_couple_opt f a b l2 lf2 -> Int_SF lf1 l1 = Int_SF lf2 l2.forall (f : R -> R) (l1 l2 lf1 lf2 : Rlist) (a b : R),
a <= b ->
adapted_couple f a b l1 lf1 ->
adapted_couple_opt f a b l2 lf2 ->
Int_SF lf1 l1 = Int_SF lf2 l2
Proof.forall (f : R -> R) (l1 l2 lf1 lf2 : Rlist) (a b : R),
a <= b ->
adapted_couple f a b l1 lf1 ->
adapted_couple_opt f a b l2 lf2 ->
Int_SF lf1 l1 = Int_SF lf2 l2
  simple induction l1.f:R -> R
l1:Rlist
forall (l2 lf1 lf2 : Rlist) (a b : R),
a <= b ->
adapted_couple f a b nil lf1 ->
adapted_couple_opt f a b l2 lf2 ->
Int_SF lf1 nil = Int_SF lf2 l2
f:R -> R
l1:Rlist
forall (r : R) (r0 : Rlist),
(forall (l2 lf1 lf2 : Rlist) (a b : R),
 a <= b ->
 adapted_couple f a b r0 lf1 ->
 adapted_couple_opt f a b l2 lf2 ->
 Int_SF lf1 r0 = Int_SF lf2 l2) ->
forall (l2 lf1 lf2 : Rlist) (a b : R),
a <= b ->
adapted_couple f a b (cons r r0) lf1 ->
adapted_couple_opt f a b l2 lf2 ->
Int_SF lf1 (cons r r0) = Int_SF lf2 l2
  intros l2 lf1 lf2 a b Hyp H H0; unfold adapted_couple in H; decompose [and] H;
    clear H H0 H2 H3 H1 H6; simpl in H4; discriminate.f:R -> R
l1:Rlist
forall (r : R) (r0 : Rlist),
(forall (l2 lf1 lf2 : Rlist) (a b : R),
 a <= b ->
 adapted_couple f a b r0 lf1 ->
 adapted_couple_opt f a b l2 lf2 ->
 Int_SF lf1 r0 = Int_SF lf2 l2) ->
forall (l2 lf1 lf2 : Rlist) (a b : R),
a <= b ->
adapted_couple f a b (cons r r0) lf1 ->
adapted_couple_opt f a b l2 lf2 ->
Int_SF lf1 (cons r r0) = Int_SF lf2 l2
  simple induction r0.f:R -> R
l1:Rlist
r:R
r0:Rlist
(forall (l2 lf1 lf2 : Rlist) (a b : R),
 a <= b ->
 adapted_couple f a b nil lf1 ->
 adapted_couple_opt f a b l2 lf2 ->
 Int_SF lf1 nil = Int_SF lf2 l2) ->
forall (l2 lf1 lf2 : Rlist) (a b : R),
a <= b ->
adapted_couple f a b (cons r nil) lf1 ->
adapted_couple_opt f a b l2 lf2 ->
Int_SF lf1 (cons r nil) = Int_SF lf2 l2
f:R -> R
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (l2 lf1 lf2 : Rlist) (a b : R),
  a <= b ->
  adapted_couple f a b r2 lf1 ->
  adapted_couple_opt f a b l2 lf2 ->
  Int_SF lf1 r2 = Int_SF lf2 l2) ->
 forall (l2 lf1 lf2 : Rlist) (a b : R),
 a <= b ->
 adapted_couple f a b (cons r r2) lf1 ->
 adapted_couple_opt f a b l2 lf2 ->
 Int_SF lf1 (cons r r2) = Int_SF lf2 l2) ->
(forall (l2 lf1 lf2 : Rlist) (a b : R),
 a <= b ->
 adapted_couple f a b (cons r1 r2) lf1 ->
 adapted_couple_opt f a b l2 lf2 ->
 Int_SF lf1 (cons r1 r2) = Int_SF lf2 l2) ->
forall (l2 lf1 lf2 : Rlist) (a b : R),
a <= b ->
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
adapted_couple_opt f a b l2 lf2 ->
Int_SF lf1 (cons r (cons r1 r2)) = Int_SF lf2 l2
  intros; case (Req_dec a b); intro.f:R -> R
l1:Rlist
r:R
r0:Rlist
H:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 nil lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 nil = Int_SF lf3 l0
l2, lf1, lf2:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r nil) lf1
H2:adapted_couple_opt f a b l2 lf2
H3:a = b
Int_SF lf1 (cons r nil) = Int_SF lf2 l2
f:R -> R
l1:Rlist
r:R
r0:Rlist
H:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 nil lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 nil = Int_SF lf3 l0
l2, lf1, lf2:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r nil) lf1
H2:adapted_couple_opt f a b l2 lf2
H3:a <> b
Int_SF lf1 (cons r nil) = Int_SF lf2 l2
f:R -> R
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (l2 lf1 lf2 : Rlist) (a b : R),
  a <= b ->
  adapted_couple f a b r2 lf1 ->
  adapted_couple_opt f a b l2 lf2 ->
  Int_SF lf1 r2 = Int_SF lf2 l2) ->
 forall (l2 lf1 lf2 : Rlist) (a b : R),
 a <= b ->
 adapted_couple f a b (cons r r2) lf1 ->
 adapted_couple_opt f a b l2 lf2 ->
 Int_SF lf1 (cons r r2) = Int_SF lf2 l2) ->
(forall (l2 lf1 lf2 : Rlist) (a b : R),
 a <= b ->
 adapted_couple f a b (cons r1 r2) lf1 ->
 adapted_couple_opt f a b l2 lf2 ->
 Int_SF lf1 (cons r1 r2) = Int_SF lf2 l2) ->
forall (l2 lf1 lf2 : Rlist) (a b : R),
a <= b ->
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
adapted_couple_opt f a b l2 lf2 ->
Int_SF lf1 (cons r (cons r1 r2)) = Int_SF lf2 l2
  unfold adapted_couple_opt in H2; elim H2; intros; rewrite (StepFun_P8 H4 H3);
    rewrite (StepFun_P8 H1 H3); reflexivity.f:R -> R
l1:Rlist
r:R
r0:Rlist
H:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 nil lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 nil = Int_SF lf3 l0
l2, lf1, lf2:Rlist
a, b:R
H0:a <= b
H1:adapted_couple f a b (cons r nil) lf1
H2:adapted_couple_opt f a b l2 lf2
H3:a <> b
Int_SF lf1 (cons r nil) = Int_SF lf2 l2
f:R -> R
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (l2 lf1 lf2 : Rlist) (a b : R),
  a <= b ->
  adapted_couple f a b r2 lf1 ->
  adapted_couple_opt f a b l2 lf2 ->
  Int_SF lf1 r2 = Int_SF lf2 l2) ->
 forall (l2 lf1 lf2 : Rlist) (a b : R),
 a <= b ->
 adapted_couple f a b (cons r r2) lf1 ->
 adapted_couple_opt f a b l2 lf2 ->
 Int_SF lf1 (cons r r2) = Int_SF lf2 l2) ->
(forall (l2 lf1 lf2 : Rlist) (a b : R),
 a <= b ->
 adapted_couple f a b (cons r1 r2) lf1 ->
 adapted_couple_opt f a b l2 lf2 ->
 Int_SF lf1 (cons r1 r2) = Int_SF lf2 l2) ->
forall (l2 lf1 lf2 : Rlist) (a b : R),
a <= b ->
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
adapted_couple_opt f a b l2 lf2 ->
Int_SF lf1 (cons r (cons r1 r2)) = Int_SF lf2 l2
  assert (H4 := StepFun_P9 H1 H3); simpl in H4;
    elim (le_Sn_O _ (le_S_n _ _ H4)).f:R -> R
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (l2 lf1 lf2 : Rlist) (a b : R),
  a <= b ->
  adapted_couple f a b r2 lf1 ->
  adapted_couple_opt f a b l2 lf2 ->
  Int_SF lf1 r2 = Int_SF lf2 l2) ->
 forall (l2 lf1 lf2 : Rlist) (a b : R),
 a <= b ->
 adapted_couple f a b (cons r r2) lf1 ->
 adapted_couple_opt f a b l2 lf2 ->
 Int_SF lf1 (cons r r2) = Int_SF lf2 l2) ->
(forall (l2 lf1 lf2 : Rlist) (a b : R),
 a <= b ->
 adapted_couple f a b (cons r1 r2) lf1 ->
 adapted_couple_opt f a b l2 lf2 ->
 Int_SF lf1 (cons r1 r2) = Int_SF lf2 l2) ->
forall (l2 lf1 lf2 : Rlist) (a b : R),
a <= b ->
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
adapted_couple_opt f a b l2 lf2 ->
Int_SF lf1 (cons r (cons r1 r2)) = Int_SF lf2 l2
  intros; clear H; unfold adapted_couple_opt in H3; elim H3; clear H3; intros;
    case (Req_dec a b); intro.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2, lf1, lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2)) lf1
H:adapted_couple f a b l2 lf2
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a = b
Int_SF lf1 (cons r (cons r1 r2)) = Int_SF lf2 l2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2, lf1, lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2)) lf1
H:adapted_couple f a b l2 lf2
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Int_SF lf1 (cons r (cons r1 r2)) = Int_SF lf2 l2
  rewrite (StepFun_P8 H2 H4); rewrite (StepFun_P8 H H4); reflexivity.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2, lf1, lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2)) lf1
H:adapted_couple f a b l2 lf2
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Int_SF lf1 (cons r (cons r1 r2)) = Int_SF lf2 l2
  assert (Hyp_min : Rmin a b = a).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2, lf1, lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2)) lf1
H:adapted_couple f a b l2 lf2
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Rmin a b = a
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2, lf1, lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2)) lf1
H:adapted_couple f a b l2 lf2
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Int_SF lf1 (cons r (cons r1 r2)) = Int_SF lf2 l2
  unfold Rmin; decide (Rle_dec a b) with H1; reflexivity.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2, lf1, lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2)) lf1
H:adapted_couple f a b l2 lf2
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Int_SF lf1 (cons r (cons r1 r2)) = Int_SF lf2 l2
  assert (Hyp_max : Rmax a b = b).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2, lf1, lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2)) lf1
H:adapted_couple f a b l2 lf2
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Rmax a b = b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2, lf1, lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2)) lf1
H:adapted_couple f a b l2 lf2
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
Int_SF lf1 (cons r (cons r1 r2)) = Int_SF lf2 l2
  unfold Rmax; decide (Rle_dec a b) with H1; reflexivity.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2, lf1, lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2)) lf1
H:adapted_couple f a b l2 lf2
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
Int_SF lf1 (cons r (cons r1 r2)) = Int_SF lf2 l2
  elim (RList_P20 _ (StepFun_P9 H H4)); intros s1 [s2 [s3 H5]]; rewrite H5 in H;
    rewrite H5; induction  lf1 as [| r3 lf1 Hreclf1].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf1 lf0 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf1 ->
adapted_couple_opt f a0 b0 l0 lf0 ->
Int_SF lf1 (cons r1 r2) = Int_SF lf0 l0
l2, lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2)) nil
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3)) lf2
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
Int_SF nil (cons r (cons r1 r2)) =
Int_SF lf2 (cons s1 (cons s2 s3))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1, lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3)) lf2
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) =
Int_SF lf2 (cons s1 (cons s2 s3))
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF lf2 (cons s1 (cons s2 s3))
  unfold adapted_couple in H2; decompose [and] H2;
    clear H H2 H4 H5 H3 H6 H8 H7 H11; simpl in H9; discriminate.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1, lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3)) lf2
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
Int_SF lf1 (cons r (cons r1 r2)) =
Int_SF lf2 (cons s1 (cons s2 s3))
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF lf2 (cons s1 (cons s2 s3))
  clear Hreclf1; induction  lf2 as [| r4 lf2 Hreclf2].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf2 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf2 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf2 l0
l2:Rlist
r3:R
lf1:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3)) nil
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength nil))%nat ->
 pos_Rl nil i <> pos_Rl nil (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl nil i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF nil (cons s1 (cons s2 s3))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
Hreclf2:adapted_couple f a b (cons s1 (cons s2 s3))
  lf2 ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i)) ->
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF lf2 (cons s1 (cons s2 s3))
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold adapted_couple in H; decompose [and] H;
    clear H H2 H4 H5 H3 H6 H8 H7 H11; simpl in H9; discriminate.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
Hreclf2:adapted_couple f a b (cons s1 (cons s2 s3))
  lf2 ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i)) ->
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF lf2 (cons s1 (cons s2 s3))
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  clear Hreclf2; assert (H6 : r = s1).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
r = s1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold adapted_couple in H, H2; decompose [and] H; decompose [and] H2;
    clear H H2; simpl in H13; simpl in H8; rewrite H13;
      rewrite H8; reflexivity.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  assert (H7 : r3 = r4 \/ r = r1).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
r3 = r4 \/ r = r1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  case (Req_dec r r1); intro.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r = r1
r3 = r4 \/ r = r1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
r3 = r4 \/ r = r1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  right; assumption.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
r3 = r4 \/ r = r1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  left; cut (r1 <= s2).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
r1 <= s2 -> r3 = r4
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
r1 <= s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  intro; unfold adapted_couple in H2, H; decompose [and] H; decompose [and] H2;
    clear H H2; set (x := (r + r1) / 2); assert (H18 := H14 0%nat);
      assert (H20 := H19 0%nat); unfold constant_D_eq, open_interval in H18, H20;
        simpl in H18; simpl in H20; rewrite <- (H18 (lt_O_Sn _) x).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
r3 = f x
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
s1 < x < s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
r1 <= s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite <- (H20 (lt_O_Sn _) x).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
f x = f x
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
r < x < r1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
s1 < x < s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
r1 <= s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  reflexivity.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
r < x < r1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
s1 < x < s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
r1 <= s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  assert (H21 := H13 0%nat (lt_O_Sn _)); simpl in H21; elim H21; intro;
    [ idtac | elim H7; assumption ]; unfold x;
      split.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
H21:r <= r1
H:r < r1
r < (r + r1) / 2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
H21:r <= r1
H:r < r1
(r + r1) / 2 < r1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
s1 < x < s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
r1 <= s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H
            | discrR ] ].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
H21:r <= r1
H:r < r1
(r + r1) / 2 < r1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
s1 < x < s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
r1 <= s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite <- (Rplus_comm r1); rewrite double;
            apply Rplus_lt_compat_l; apply H
            | discrR ] ].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
s1 < x < s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
r1 <= s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite <- H6; assert (H21 := H13 0%nat (lt_O_Sn _)); simpl in H21; elim H21;
    intro; [ idtac | elim H7; assumption ]; unfold x;
      split.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
H21:r <= r1
H:r < r1
r < (r + r1) / 2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
H21:r <= r1
H:r < r1
(r + r1) / 2 < s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
r1 <= s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H
            | discrR ] ].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
H8:r1 <= s2
H9:ordered_Rlist (cons s1 (cons s2 s3))
H11:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H10:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H12:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H14:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H13:ordered_Rlist (cons r (cons r1 r2))
H16:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H15:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H17:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H19:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
x:=(r + r1) / 2:R
H18:(0 < S (Rlength s3))%nat ->
forall x0 : R, s1 < x0 < s2 -> f x0 = r4
H20:(0 < S (Rlength r2))%nat ->
forall x0 : R, r < x0 < r1 -> f x0 = r3
H21:r <= r1
H:r < r1
(r + r1) / 2 < s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
r1 <= s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply Rlt_le_trans with r1;
    [ apply Rmult_lt_reg_l with 2;
      [ prove_sup0
        | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
          rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
            [ rewrite Rmult_1_l; rewrite <- (Rplus_comm r1); rewrite double;
              apply Rplus_lt_compat_l; apply H
              | discrR ] ]
      | assumption ].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
r1 <= s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  eapply StepFun_P13.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
?a <> ?b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
adapted_couple ?f ?a ?b 
  (cons ?r (cons r1 ?r2)) 
  (cons ?r3 ?lf1)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
adapted_couple_opt ?f ?a 
  ?b (cons ?s1 (cons s2 ?s3)) 
  (cons ?r4 ?lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply H4.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
adapted_couple ?f a b (cons ?r (cons r1 ?r2))
  (cons ?r3 ?lf1)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
adapted_couple_opt ?f a b 
  (cons ?s1 (cons s2 ?s3)) 
  (cons ?r4 ?lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply H2.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
adapted_couple_opt f a b 
  (cons ?s1 (cons s2 ?s3)) 
  (cons ?r4 ?lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold adapted_couple_opt; split.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
adapted_couple f a b (cons ?s1 (cons s2 ?s3))
  (cons ?r4 ?lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons ?r4 ?lf2)))%nat ->
 pos_Rl (cons ?r4 ?lf2) i <>
 pos_Rl (cons ?r4 ?lf2) (S i) \/
 f (pos_Rl (cons ?s1 (cons s2 ?s3)) (S i)) <>
 pos_Rl (cons ?r4 ?lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons ?s1 (cons s2 ?s3))))%nat ->
 pos_Rl (cons ?s1 (cons s2 ?s3)) i <>
 pos_Rl (cons ?s1 (cons s2 ?s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply H.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r <> r1
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <> pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite H5 in H3; apply H3.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  assert (H8 : r1 <= s2).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
r1 <= s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  eapply StepFun_P13.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
?a <> ?b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
adapted_couple ?f ?a ?b 
  (cons ?r (cons r1 ?r2)) 
  (cons ?r3 ?lf1)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
adapted_couple_opt ?f ?a 
  ?b (cons ?s1 (cons s2 ?s3)) 
  (cons ?r4 ?lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply H4.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
adapted_couple ?f a b (cons ?r (cons r1 ?r2))
  (cons ?r3 ?lf1)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
adapted_couple_opt ?f a b 
  (cons ?s1 (cons s2 ?s3)) 
  (cons ?r4 ?lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply H2.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
adapted_couple_opt f a b 
  (cons ?s1 (cons s2 ?s3)) 
  (cons ?r4 ?lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold adapted_couple_opt; split.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
adapted_couple f a b (cons ?s1 (cons s2 ?s3))
  (cons ?r4 ?lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons ?r4 ?lf2)))%nat ->
 pos_Rl (cons ?r4 ?lf2) i <>
 pos_Rl (cons ?r4 ?lf2) (S i) \/
 f (pos_Rl (cons ?s1 (cons s2 ?s3)) (S i)) <>
 pos_Rl (cons ?r4 ?lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons ?s1 (cons s2 ?s3))))%nat ->
 pos_Rl (cons ?s1 (cons s2 ?s3)) i <>
 pos_Rl (cons ?s1 (cons s2 ?s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply H.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <> pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
 pos_Rl (cons s1 (cons s2 s3)) i <>
 pos_Rl (cons s1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite H5 in H3; apply H3.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  elim H7; intro.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; elim H8; intro.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  replace (r4 * (s2 - s1)) with (r3 * (r1 - r) + r3 * (s2 - r1));
  [ idtac | rewrite H9; rewrite H6; ring ].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r3 * (r1 - r) + r3 * (s2 - r1) +
Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite Rplus_assoc; apply Rplus_eq_compat_l;
    change
      (Int_SF lf1 (cons r1 r2) = Int_SF (cons r3 lf2) (cons r1 (cons s2 s3)))
     ; apply H0 with r1 b.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
r1 <= b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple f r1 b (cons r1 r2) lf1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple_opt f r1 b 
  (cons r1 (cons s2 s3)) 
  (cons r3 lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold adapted_couple in H2; decompose [and] H2; clear H2;
    replace b with (Rmax a b).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
r1 <= Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple f r1 b (cons r1 r2) lf1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple_opt f r1 b 
  (cons r1 (cons s2 s3)) 
  (cons r3 lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite <- H12; apply RList_P7;
    [ assumption | simpl; right; left; reflexivity ].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple f r1 b (cons r1 r2) lf1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple_opt f r1 b 
  (cons r1 (cons s2 s3)) 
  (cons r3 lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  eapply StepFun_P7.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
?a <= b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple f ?a b (cons ?r1 (cons r1 r2))
  (cons ?r3 lf1)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple_opt f r1 b 
  (cons r1 (cons s2 s3)) 
  (cons r3 lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply H1.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple f a b (cons ?r1 (cons r1 r2))
  (cons ?r3 lf1)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple_opt f r1 b 
  (cons r1 (cons s2 s3)) 
  (cons r3 lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply H2.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple_opt f r1 b 
  (cons r1 (cons s2 s3)) 
  (cons r3 lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold adapted_couple_opt; split.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple f r1 b (cons r1 (cons s2 s3))
  (cons r3 lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply StepFun_P7 with a a r3.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
a <= b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple f a b (cons a (cons r1 (cons s2 s3)))
  (cons r3 (cons r3 lf2))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply H1.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
adapted_couple f a b (cons a (cons r1 (cons s2 s3)))
  (cons r3 (cons r3 lf2))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold adapted_couple in H2, H; decompose [and] H2; decompose [and] H;
    clear H H2; assert (H20 : r = a).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
r = a
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
adapted_couple f a b (cons a (cons r1 (cons s2 s3)))
  (cons r3 (cons r3 lf2))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl in H13; rewrite H13; apply Hyp_min.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
adapted_couple f a b (cons a (cons r1 (cons s2 s3)))
  (cons r3 (cons r3 lf2))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold adapted_couple; repeat split.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
ordered_Rlist (cons a (cons r1 (cons s2 s3)))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3))) 0 = Rmin a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (Init.Nat.pred
     (Rlength (cons a (cons r1 (cons s2 s3))))) =
Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 (cons s2 s3))) =
S (Rlength (cons r3 (cons r3 lf2)))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons a (cons r1 (cons s2 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons a (cons r1 (cons s2 s3))) i)
     (pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)))
  (pos_Rl (cons r3 (cons r3 lf2)) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold ordered_Rlist; intros; simpl in H; induction  i as [| i Hreci].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(0 < S (S (Rlength s3)))%nat
pos_Rl (cons a (cons r1 (cons s2 s3))) 0 <=
pos_Rl (cons a (cons r1 (cons s2 s3))) 1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (S (Rlength s3)))%nat
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) i <=
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <=
pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3))) 0 = Rmin a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (Init.Nat.pred
     (Rlength (cons a (cons r1 (cons s2 s3))))) =
Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 (cons s2 s3))) =
S (Rlength (cons r3 (cons r3 lf2)))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons a (cons r1 (cons s2 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons a (cons r1 (cons s2 s3))) i)
     (pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)))
  (pos_Rl (cons r3 (cons r3 lf2)) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; rewrite <- H20; apply (H11 0%nat).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(0 < S (S (Rlength s3)))%nat
(0 < Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (S (Rlength s3)))%nat
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) i <=
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <=
pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3))) 0 = Rmin a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (Init.Nat.pred
     (Rlength (cons a (cons r1 (cons s2 s3))))) =
Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 (cons s2 s3))) =
S (Rlength (cons r3 (cons r3 lf2)))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons a (cons r1 (cons s2 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons a (cons r1 (cons s2 s3))) i)
     (pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)))
  (pos_Rl (cons r3 (cons r3 lf2)) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; apply lt_O_Sn.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (S (Rlength s3)))%nat
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) i <=
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <=
pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3))) 0 = Rmin a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (Init.Nat.pred
     (Rlength (cons a (cons r1 (cons s2 s3))))) =
Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 (cons s2 s3))) =
S (Rlength (cons r3 (cons r3 lf2)))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons a (cons r1 (cons s2 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons a (cons r1 (cons s2 s3))) i)
     (pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)))
  (pos_Rl (cons r3 (cons r3 lf2)) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  induction  i as [| i Hreci0].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(1 < S (S (Rlength s3)))%nat
Hreci:(0 < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) 0 <=
pos_Rl (cons a (cons r1 (cons s2 s3))) 1
pos_Rl (cons a (cons r1 (cons s2 s3))) 1 <=
pos_Rl (cons a (cons r1 (cons s2 s3))) 2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S (S i) < S (S (Rlength s3)))%nat
Hreci:(S i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <=
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S i))
Hreci0:(S i < S (S (Rlength s3)))%nat ->
((i < S (S (Rlength s3)))%nat ->
 pos_Rl (cons a (cons r1 (cons s2 s3))) i <=
 pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)) ->
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <=
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S i))
pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i)) <=
pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S (S i)))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3))) 0 = Rmin a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (Init.Nat.pred
     (Rlength (cons a (cons r1 (cons s2 s3))))) =
Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 (cons s2 s3))) =
S (Rlength (cons r3 (cons r3 lf2)))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons a (cons r1 (cons s2 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons a (cons r1 (cons s2 s3))) i)
     (pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)))
  (pos_Rl (cons r3 (cons r3 lf2)) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; assumption.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S (S i) < S (S (Rlength s3)))%nat
Hreci:(S i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <=
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S i))
Hreci0:(S i < S (S (Rlength s3)))%nat ->
((i < S (S (Rlength s3)))%nat ->
 pos_Rl (cons a (cons r1 (cons s2 s3))) i <=
 pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)) ->
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <=
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S i))
pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i)) <=
pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S (S i)))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3))) 0 = Rmin a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (Init.Nat.pred
     (Rlength (cons a (cons r1 (cons s2 s3))))) =
Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 (cons s2 s3))) =
S (Rlength (cons r3 (cons r3 lf2)))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons a (cons r1 (cons s2 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons a (cons r1 (cons s2 s3))) i)
     (pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)))
  (pos_Rl (cons r3 (cons r3 lf2)) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  change (pos_Rl (cons s2 s3) i <= pos_Rl (cons s2 s3) (S i));
    apply (H15 (S i)); simpl; apply lt_S_n; assumption.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3))) 0 = Rmin a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (Init.Nat.pred
     (Rlength (cons a (cons r1 (cons s2 s3))))) =
Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 (cons s2 s3))) =
S (Rlength (cons r3 (cons r3 lf2)))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons a (cons r1 (cons s2 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons a (cons r1 (cons s2 s3))) i)
     (pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)))
  (pos_Rl (cons r3 (cons r3 lf2)) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; symmetry ; apply Hyp_min.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (Init.Nat.pred
     (Rlength (cons a (cons r1 (cons s2 s3))))) =
Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 (cons s2 s3))) =
S (Rlength (cons r3 (cons r3 lf2)))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons a (cons r1 (cons s2 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons a (cons r1 (cons s2 s3))) i)
     (pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)))
  (pos_Rl (cons r3 (cons r3 lf2)) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite <- H17; reflexivity.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 (cons s2 s3))) =
S (Rlength (cons r3 (cons r3 lf2)))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons a (cons r1 (cons s2 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons a (cons r1 (cons s2 s3))) i)
     (pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)))
  (pos_Rl (cons r3 (cons r3 lf2)) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl in H19; simpl; rewrite H19; reflexivity.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i <
 Init.Nat.pred
   (Rlength (cons a (cons r1 (cons s2 s3)))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons a (cons r1 (cons s2 s3))) i)
     (pos_Rl (cons a (cons r1 (cons s2 s3))) (S i)))
  (pos_Rl (cons r3 (cons r3 lf2)) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  intros; simpl in H; unfold constant_D_eq, open_interval; intros;
    induction  i as [| i Hreci].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(0 < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) 0 < x <
pos_Rl (cons a (cons r1 (cons s2 s3))) 1
f x = pos_Rl (cons r3 (cons r3 lf2)) 0
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i))
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) i < x <
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) ->
f x = pos_Rl (cons r3 (cons r3 lf2)) i
f x = pos_Rl (cons r3 (cons r3 lf2)) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; apply (H16 0%nat).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(0 < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) 0 < x <
pos_Rl (cons a (cons r1 (cons s2 s3))) 1
(0 < Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(0 < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) 0 < x <
pos_Rl (cons a (cons r1 (cons s2 s3))) 1
open_interval (pos_Rl (cons r (cons r1 r2)) 0)
  (pos_Rl (cons r (cons r1 r2)) 1) x
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i))
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) i < x <
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) ->
f x = pos_Rl (cons r3 (cons r3 lf2)) i
f x = pos_Rl (cons r3 (cons r3 lf2)) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; apply lt_O_Sn.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(0 < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) 0 < x <
pos_Rl (cons a (cons r1 (cons s2 s3))) 1
open_interval (pos_Rl (cons r (cons r1 r2)) 0)
  (pos_Rl (cons r (cons r1 r2)) 1) x
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i))
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) i < x <
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) ->
f x = pos_Rl (cons r3 (cons r3 lf2)) i
f x = pos_Rl (cons r3 (cons r3 lf2)) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl in H2; rewrite <- H20 in H2; unfold open_interval;
    simpl; apply H2.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i))
Hreci:(i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) i < x <
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) ->
f x = pos_Rl (cons r3 (cons r3 lf2)) i
f x = pos_Rl (cons r3 (cons r3 lf2)) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  clear Hreci; induction  i as [| i Hreci].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(1 < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) 1 < x <
pos_Rl (cons a (cons r1 (cons s2 s3))) 2
f x = pos_Rl (cons r3 (cons r3 lf2)) 1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S (S i) < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i)) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S (S i)))
Hreci:(S i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S i)) ->
f x = pos_Rl (cons r3 (cons r3 lf2)) (S i)
f x = pos_Rl (cons r3 (cons r3 lf2)) (S (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; simpl in H2; rewrite H9; apply (H21 0%nat).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(1 < S (S (Rlength s3)))%nat
x:R
H2:r1 < x < s2
(0 < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(1 < S (S (Rlength s3)))%nat
x:R
H2:r1 < x < s2
open_interval (pos_Rl (cons s1 (cons s2 s3)) 0)
  (pos_Rl (cons s1 (cons s2 s3)) 1) x
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S (S i) < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i)) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S (S i)))
Hreci:(S i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S i)) ->
f x = pos_Rl (cons r3 (cons r3 lf2)) (S i)
f x = pos_Rl (cons r3 (cons r3 lf2)) (S (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; apply lt_O_Sn.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(1 < S (S (Rlength s3)))%nat
x:R
H2:r1 < x < s2
open_interval (pos_Rl (cons s1 (cons s2 s3)) 0)
  (pos_Rl (cons s1 (cons s2 s3)) 1) x
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S (S i) < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i)) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S (S i)))
Hreci:(S i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S i)) ->
f x = pos_Rl (cons r3 (cons r3 lf2)) (S i)
f x = pos_Rl (cons r3 (cons r3 lf2)) (S (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold open_interval; simpl; elim H2; intros; split.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(1 < S (S (Rlength s3)))%nat
x:R
H2:r1 < x < s2
H22:r1 < x
H23:x < s2
s1 < x
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(1 < S (S (Rlength s3)))%nat
x:R
H2:r1 < x < s2
H22:r1 < x
H23:x < s2
x < s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S (S i) < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i)) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S (S i)))
Hreci:(S i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S i)) ->
f x = pos_Rl (cons r3 (cons r3 lf2)) (S i)
f x = pos_Rl (cons r3 (cons r3 lf2)) (S (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply Rle_lt_trans with r1; try assumption; rewrite <- H6; apply (H11 0%nat);
    simpl; apply lt_O_Sn.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(1 < S (S (Rlength s3)))%nat
x:R
H2:r1 < x < s2
H22:r1 < x
H23:x < s2
x < s2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S (S i) < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i)) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S (S i)))
Hreci:(S i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S i)) ->
f x = pos_Rl (cons r3 (cons r3 lf2)) (S i)
f x = pos_Rl (cons r3 (cons r3 lf2)) (S (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  assumption.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S (S i) < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i)) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S (S i)))
Hreci:(S i < S (S (Rlength s3)))%nat ->
pos_Rl (cons a (cons r1 (cons s2 s3))) (S i) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S i)) ->
f x = pos_Rl (cons r3 (cons r3 lf2)) (S i)
f x = pos_Rl (cons r3 (cons r3 lf2)) (S (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  clear Hreci; simpl; apply (H21 (S i)).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S (S i) < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i)) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S (S i)))
(S i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S (S i) < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i)) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S (S i)))
open_interval (pos_Rl (cons s1 (cons s2 s3)) (S i))
  (pos_Rl (cons s1 (cons s2 s3)) (S (S i))) x
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; apply lt_S_n; assumption.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S (S i) < S (S (Rlength s3)))%nat
x:R
H2:pos_Rl (cons a (cons r1 (cons s2 s3))) (S (S i)) <
x <
pos_Rl (cons a (cons r1 (cons s2 s3)))
  (S (S (S i)))
open_interval (pos_Rl (cons s1 (cons s2 s3)) (S i))
  (pos_Rl (cons s1 (cons s2 s3)) (S (S i))) x
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold open_interval; apply H2.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
 pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
 f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
 pos_Rl (cons r3 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
 pos_Rl (cons r1 (cons s2 s3)) i <>
 pos_Rl (cons r1 (cons s2 s3)) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  elim H3; clear H3; intros; split.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H3:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i <>
pos_Rl (cons r4 lf2) (S i) \/
f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i
H11:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i <> pos_Rl l2 (S i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r3 lf2)))%nat ->
pos_Rl (cons r3 lf2) i <> pos_Rl (cons r3 lf2) (S i) \/
f (pos_Rl (cons r1 (cons s2 s3)) (S i)) <>
pos_Rl (cons r3 lf2) i
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H3:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i <>
pos_Rl (cons r4 lf2) (S i) \/
f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i
H11:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i <> pos_Rl l2 (S i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
pos_Rl (cons r1 (cons s2 s3)) i <>
pos_Rl (cons r1 (cons s2 s3)) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite H9;
    change
      (forall i:nat,
        (i < pred (Rlength (cons r4 lf2)))%nat ->
        pos_Rl (cons r4 lf2) i <> pos_Rl (cons r4 lf2) (S i) \/
        f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <> pos_Rl (cons r4 lf2) i)
     ; rewrite <- H5; apply H3.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H3:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i <>
pos_Rl (cons r4 lf2) (S i) \/
f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i
H11:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i <> pos_Rl l2 (S i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 (cons s2 s3))))%nat ->
pos_Rl (cons r1 (cons s2 s3)) i <>
pos_Rl (cons r1 (cons s2 s3)) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite H5 in H11; intros; simpl in H12; induction  i as [| i Hreci].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H3:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i <>
pos_Rl (cons r4 lf2) (S i) \/
f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i
H11:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
pos_Rl (cons s1 (cons s2 s3)) i <>
pos_Rl (cons s1 (cons s2 s3)) (S i)
H12:(0 < S (Rlength s3))%nat
pos_Rl (cons r1 (cons s2 s3)) 0 <>
pos_Rl (cons r1 (cons s2 s3)) 1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H3:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i0 <>
pos_Rl (cons r4 lf2) (S i0) \/
f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0
H11:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
pos_Rl (cons s1 (cons s2 s3)) i0 <>
pos_Rl (cons s1 (cons s2 s3)) (S i0)
i:nat
H12:(S i < S (Rlength s3))%nat
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons r1 (cons s2 s3)) i <>
pos_Rl (cons r1 (cons s2 s3)) (S i)
pos_Rl (cons r1 (cons s2 s3)) (S i) <>
pos_Rl (cons r1 (cons s2 s3)) (S (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; red; intro; rewrite H13 in H10;
    elim (Rlt_irrefl _ H10).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 < s2
H3:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i0 <>
pos_Rl (cons r4 lf2) (S i0) \/
f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0
H11:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
pos_Rl (cons s1 (cons s2 s3)) i0 <>
pos_Rl (cons s1 (cons s2 s3)) (S i0)
i:nat
H12:(S i < S (Rlength s3))%nat
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons r1 (cons s2 s3)) i <>
pos_Rl (cons r1 (cons s2 s3)) (S i)
pos_Rl (cons r1 (cons s2 s3)) (S i) <>
pos_Rl (cons r1 (cons s2 s3)) (S (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  clear Hreci; apply (H11 (S i)); simpl; apply H12.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r3 * (r1 - r) + Int_SF lf1 (cons r1 r2) =
r4 * (s2 - s1) + Int_SF lf2 (cons s2 s3)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite H9; rewrite H10; rewrite H6; apply Rplus_eq_compat_l; rewrite <- H10;
    apply H0 with r1 b.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
r1 <= b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple f r1 b (cons r1 r2) lf1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple_opt f r1 b (cons r1 s3) lf2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold adapted_couple in H2; decompose [and] H2; clear H2;
    replace b with (Rmax a b).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
r1 <= Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple f r1 b (cons r1 r2) lf1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple_opt f r1 b (cons r1 s3) lf2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite <- H12; apply RList_P7;
    [ assumption | simpl; right; left; reflexivity ].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple f r1 b (cons r1 r2) lf1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple_opt f r1 b (cons r1 s3) lf2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  eapply StepFun_P7.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
?a <= b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple f ?a b (cons ?r1 (cons r1 r2))
  (cons ?r3 lf1)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple_opt f r1 b (cons r1 s3) lf2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply H1.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple f a b (cons ?r1 (cons r1 r2))
  (cons ?r3 lf1)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple_opt f r1 b (cons r1 s3) lf2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply H2.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple_opt f r1 b (cons r1 s3) lf2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold adapted_couple_opt; split.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple f r1 b (cons r1 s3) lf2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply StepFun_P7 with a a r3.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
a <= b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple f a b (cons a (cons r1 s3))
  (cons r3 lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply H1.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
adapted_couple f a b (cons a (cons r1 s3))
  (cons r3 lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold adapted_couple in H2, H; decompose [and] H2; decompose [and] H;
    clear H H2; assert (H20 : r = a).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
r = a
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
adapted_couple f a b (cons a (cons r1 s3))
  (cons r3 lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl in H13; rewrite H13; apply Hyp_min.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
adapted_couple f a b (cons a (cons r1 s3))
  (cons r3 lf2)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold adapted_couple; repeat split.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
ordered_Rlist (cons a (cons r1 s3))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 s3)) 0 = Rmin a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 s3))
  (Init.Nat.pred (Rlength (cons a (cons r1 s3)))) =
Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 s3)) =
S (Rlength (cons r3 lf2))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons r1 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons r1 s3)) i)
     (pos_Rl (cons a (cons r1 s3)) (S i)))
  (pos_Rl (cons r3 lf2) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  unfold ordered_Rlist; intros; simpl in H; induction  i as [| i Hreci].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(0 < S (Rlength s3))%nat
pos_Rl (cons a (cons r1 s3)) 0 <=
pos_Rl (cons a (cons r1 s3)) 1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (Rlength s3))%nat
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons a (cons r1 s3)) i <=
pos_Rl (cons a (cons r1 s3)) (S i)
pos_Rl (cons a (cons r1 s3)) (S i) <=
pos_Rl (cons a (cons r1 s3)) (S (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 s3)) 0 = Rmin a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 s3))
  (Init.Nat.pred (Rlength (cons a (cons r1 s3)))) =
Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 s3)) =
S (Rlength (cons r3 lf2))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons r1 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons r1 s3)) i)
     (pos_Rl (cons a (cons r1 s3)) (S i)))
  (pos_Rl (cons r3 lf2) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; rewrite <- H20; apply (H11 0%nat); simpl;
    apply lt_O_Sn.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (Rlength s3))%nat
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons a (cons r1 s3)) i <=
pos_Rl (cons a (cons r1 s3)) (S i)
pos_Rl (cons a (cons r1 s3)) (S i) <=
pos_Rl (cons a (cons r1 s3)) (S (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 s3)) 0 = Rmin a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 s3))
  (Init.Nat.pred (Rlength (cons a (cons r1 s3)))) =
Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 s3)) =
S (Rlength (cons r3 lf2))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons r1 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons r1 s3)) i)
     (pos_Rl (cons a (cons r1 s3)) (S i)))
  (pos_Rl (cons r3 lf2) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite H10; apply (H15 (S i)); simpl; assumption.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 s3)) 0 = Rmin a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 s3))
  (Init.Nat.pred (Rlength (cons a (cons r1 s3)))) =
Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 s3)) =
S (Rlength (cons r3 lf2))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons r1 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons r1 s3)) i)
     (pos_Rl (cons a (cons r1 s3)) (S i)))
  (pos_Rl (cons r3 lf2) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; symmetry ; apply Hyp_min.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
pos_Rl (cons a (cons r1 s3))
  (Init.Nat.pred (Rlength (cons a (cons r1 s3)))) =
Rmax a b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 s3)) =
S (Rlength (cons r3 lf2))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons r1 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons r1 s3)) i)
     (pos_Rl (cons a (cons r1 s3)) (S i)))
  (pos_Rl (cons r3 lf2) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite <- H17; rewrite H10; reflexivity.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
Rlength (cons a (cons r1 s3)) =
S (Rlength (cons r3 lf2))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons r1 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons r1 s3)) i)
     (pos_Rl (cons a (cons r1 s3)) (S i)))
  (pos_Rl (cons r3 lf2) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl in H19; simpl; apply H19.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a (cons r1 s3))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons a (cons r1 s3)) i)
     (pos_Rl (cons a (cons r1 s3)) (S i)))
  (pos_Rl (cons r3 lf2) i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  intros; simpl in H; unfold constant_D_eq, open_interval; intros;
    induction  i as [| i Hreci].f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(0 < S (Rlength s3))%nat
x:R
H2:pos_Rl (cons a (cons r1 s3)) 0 < x <
pos_Rl (cons a (cons r1 s3)) 1
f x = pos_Rl (cons r3 lf2) 0
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (Rlength s3))%nat
x:R
H2:pos_Rl (cons a (cons r1 s3)) (S i) < x <
pos_Rl (cons a (cons r1 s3)) (S (S i))
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons a (cons r1 s3)) i < x <
pos_Rl (cons a (cons r1 s3)) (S i) ->
f x = pos_Rl (cons r3 lf2) i
f x = pos_Rl (cons r3 lf2) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; apply (H16 0%nat).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(0 < S (Rlength s3))%nat
x:R
H2:pos_Rl (cons a (cons r1 s3)) 0 < x <
pos_Rl (cons a (cons r1 s3)) 1
(0 < Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(0 < S (Rlength s3))%nat
x:R
H2:pos_Rl (cons a (cons r1 s3)) 0 < x <
pos_Rl (cons a (cons r1 s3)) 1
open_interval (pos_Rl (cons r (cons r1 r2)) 0)
  (pos_Rl (cons r (cons r1 r2)) 1) x
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (Rlength s3))%nat
x:R
H2:pos_Rl (cons a (cons r1 s3)) (S i) < x <
pos_Rl (cons a (cons r1 s3)) (S (S i))
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons a (cons r1 s3)) i < x <
pos_Rl (cons a (cons r1 s3)) (S i) ->
f x = pos_Rl (cons r3 lf2) i
f x = pos_Rl (cons r3 lf2) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; apply lt_O_Sn.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i)
     (pos_Rl (cons s1 (cons s2 s3)) (S i)))
  (pos_Rl (cons r4 lf2) i)
H20:r = a
H:(0 < S (Rlength s3))%nat
x:R
H2:pos_Rl (cons a (cons r1 s3)) 0 < x <
pos_Rl (cons a (cons r1 s3)) 1
open_interval (pos_Rl (cons r (cons r1 r2)) 0)
  (pos_Rl (cons r (cons r1 r2)) 1) x
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (Rlength s3))%nat
x:R
H2:pos_Rl (cons a (cons r1 s3)) (S i) < x <
pos_Rl (cons a (cons r1 s3)) (S (S i))
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons a (cons r1 s3)) i < x <
pos_Rl (cons a (cons r1 s3)) (S i) ->
f x = pos_Rl (cons r3 lf2) i
f x = pos_Rl (cons r3 lf2) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl in H2; rewrite <- H20 in H2; unfold open_interval;
    simpl; apply H2.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (Rlength s3))%nat
x:R
H2:pos_Rl (cons a (cons r1 s3)) (S i) < x <
pos_Rl (cons a (cons r1 s3)) (S (S i))
Hreci:(i < S (Rlength s3))%nat ->
pos_Rl (cons a (cons r1 s3)) i < x <
pos_Rl (cons a (cons r1 s3)) (S i) ->
f x = pos_Rl (cons r3 lf2) i
f x = pos_Rl (cons r3 lf2) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  clear Hreci; simpl; apply (H21 (S i)).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (Rlength s3))%nat
x:R
H2:pos_Rl (cons a (cons r1 s3)) (S i) < x <
pos_Rl (cons a (cons r1 s3)) (S (S i))
(S i < Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (Rlength s3))%nat
x:R
H2:pos_Rl (cons a (cons r1 s3)) (S i) < x <
pos_Rl (cons a (cons r1 s3)) (S (S i))
open_interval (pos_Rl (cons s1 (cons s2 s3)) (S i))
  (pos_Rl (cons s1 (cons s2 s3)) (S (S i))) x
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; assumption.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
s1, s2:R
s3:Rlist
H3:(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i0 <>
 pos_Rl (cons r4 lf2) (S i0) \/
 f (pos_Rl l2 (S i0)) <> pos_Rl (cons r4 lf2) i0) /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i0 <> pos_Rl l2 (S i0))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H11:ordered_Rlist (cons r (cons r1 r2))
H13:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H12:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H14:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H16:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H15:ordered_Rlist (cons s1 (cons s2 s3))
H18:pos_Rl (cons s1 (cons s2 s3)) 0 = Rmin a b
H17:pos_Rl (cons s1 (cons s2 s3))
  (Init.Nat.pred
     (Rlength (cons s1 (cons s2 s3)))) =
Rmax a b
H19:Rlength (cons s1 (cons s2 s3)) =
S (Rlength (cons r4 lf2))
H21:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons s1 (cons s2 s3))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons s1 (cons s2 s3)) i0)
     (pos_Rl (cons s1 (cons s2 s3)) (S i0)))
  (pos_Rl (cons r4 lf2) i0)
H20:r = a
i:nat
H:(S i < S (Rlength s3))%nat
x:R
H2:pos_Rl (cons a (cons r1 s3)) (S i) < x <
pos_Rl (cons a (cons r1 s3)) (S (S i))
open_interval (pos_Rl (cons s1 (cons s2 s3)) (S i))
  (pos_Rl (cons s1 (cons s2 s3)) (S (S i))) x
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite <- H10; unfold open_interval; apply H2.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf2))%nat ->
 pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
 f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
 pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i))
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  elim H3; clear H3; intros; split.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H3:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i <>
pos_Rl (cons r4 lf2) (S i) \/
f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i
H11:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i <> pos_Rl l2 (S i)
forall i : nat,
(i < Init.Nat.pred (Rlength lf2))%nat ->
pos_Rl lf2 i <> pos_Rl lf2 (S i) \/
f (pos_Rl (cons r1 s3) (S i)) <> pos_Rl lf2 i
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H3:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i <>
pos_Rl (cons r4 lf2) (S i) \/
f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i
H11:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i <> pos_Rl l2 (S i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  rewrite H5 in H3; intros; apply (H3 (S i)).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H3:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i0 <>
pos_Rl (cons r4 lf2) (S i0) \/
f (pos_Rl (cons s1 (cons s2 s3)) (S i0)) <>
pos_Rl (cons r4 lf2) i0
H11:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i0 <> pos_Rl l2 (S i0)
i:nat
H12:(i < Init.Nat.pred (Rlength lf2))%nat
(S i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H3:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i <>
pos_Rl (cons r4 lf2) (S i) \/
f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i
H11:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i <> pos_Rl l2 (S i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; replace (Rlength lf2) with (S (pred (Rlength lf2))).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H3:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i0 <>
pos_Rl (cons r4 lf2) (S i0) \/
f (pos_Rl (cons s1 (cons s2 s3)) (S i0)) <>
pos_Rl (cons r4 lf2) i0
H11:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i0 <> pos_Rl l2 (S i0)
i:nat
H12:(i < Init.Nat.pred (Rlength lf2))%nat
(S i < S (Init.Nat.pred (Rlength lf2)))%nat
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H3:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i0 <>
pos_Rl (cons r4 lf2) (S i0) \/
f (pos_Rl (cons s1 (cons s2 s3)) (S i0)) <>
pos_Rl (cons r4 lf2) i0
H11:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i0 <> pos_Rl l2 (S i0)
i:nat
H12:(i < Init.Nat.pred (Rlength lf2))%nat
S (Init.Nat.pred (Rlength lf2)) = Rlength lf2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H3:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i <>
pos_Rl (cons r4 lf2) (S i) \/
f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i
H11:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i <> pos_Rl l2 (S i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  apply lt_n_S; apply H12.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H3:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i0 <>
pos_Rl (cons r4 lf2) (S i0) \/
f (pos_Rl (cons s1 (cons s2 s3)) (S i0)) <>
pos_Rl (cons r4 lf2) i0
H11:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i0 <> pos_Rl l2 (S i0)
i:nat
H12:(i < Init.Nat.pred (Rlength lf2))%nat
S (Init.Nat.pred (Rlength lf2)) = Rlength lf2
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H3:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i <>
pos_Rl (cons r4 lf2) (S i) \/
f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i
H11:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i <> pos_Rl l2 (S i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red;
    intro; rewrite <- H13 in H12; elim (lt_n_O _ H12).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r3 = r4
H10:r1 = s2
H3:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i <>
pos_Rl (cons r4 lf2) (S i) \/
f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i
H11:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i <> pos_Rl l2 (S i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 s3)))%nat ->
pos_Rl (cons r1 s3) i <> pos_Rl (cons r1 s3) (S i)
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  intros; simpl in H12; rewrite H10; rewrite H5 in H11; apply (H11 (S i));
    simpl; apply lt_n_S; apply H12.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
Int_SF (cons r3 lf1) (cons r (cons r1 r2)) =
Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))
  simpl; rewrite H9; unfold Rminus; rewrite Rplus_opp_r;
    rewrite Rmult_0_r; rewrite Rplus_0_l;
      change
        (Int_SF lf1 (cons r1 r2) = Int_SF (cons r4 lf2) (cons s1 (cons s2 s3)))
       ; eapply H0.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
?a <= ?b
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
adapted_couple f ?a ?b (cons r1 r2) lf1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
adapted_couple_opt f ?a 
  ?b (cons s1 (cons s2 s3)) 
  (cons r4 lf2)
  apply H1.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
adapted_couple f a b (cons r1 r2) lf1
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
adapted_couple_opt f a b 
  (cons s1 (cons s2 s3)) 
  (cons r4 lf2)
  2: rewrite H5 in H3; unfold adapted_couple_opt; split; assumption.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
adapted_couple f a b (cons r1 r2) lf1
  assert (H10 : r = a).f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
r = a
f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
H10:r = a
adapted_couple f a b (cons r1 r2) lf1
  unfold adapted_couple in H2; decompose [and] H2; clear H2; simpl in H12;
    rewrite H12; apply Hyp_min.f:R -> R
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
H0:forall (l0 lf0 lf3 : Rlist) (a0 b0 : R),
a0 <= b0 ->
adapted_couple f a0 b0 (cons r1 r2) lf0 ->
adapted_couple_opt f a0 b0 l0 lf3 ->
Int_SF lf0 (cons r1 r2) = Int_SF lf3 l0
l2:Rlist
r3:R
lf1:Rlist
r4:R
lf2:Rlist
a, b:R
H1:a <= b
H2:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
s1, s2:R
s3:Rlist
H:adapted_couple f a b (cons s1 (cons s2 s3))
  (cons r4 lf2)
H3:(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r4 lf2)))%nat ->
 pos_Rl (cons r4 lf2) i <>
 pos_Rl (cons r4 lf2) (S i) \/
 f (pos_Rl l2 (S i)) <> pos_Rl (cons r4 lf2) i) /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l2))%nat ->
 pos_Rl l2 i <> pos_Rl l2 (S i))
H4:a <> b
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H5:l2 = cons s1 (cons s2 s3)
H6:r = s1
H7:r3 = r4 \/ r = r1
H8:r1 <= s2
H9:r = r1
H10:r = a
adapted_couple f a b (cons r1 r2) lf1
  rewrite <- H9; rewrite H10; apply StepFun_P7 with a r r3;
    [ apply H1
      | pattern a at 2; rewrite <- H10; pattern r at 2; rewrite H9;
        apply H2 ].
Qed.

Lemma StepFun_P15 :
  forall (f:R -> R) (l1 l2 lf1 lf2:Rlist) (a b:R),
    adapted_couple f a b l1 lf1 ->
    adapted_couple_opt f a b l2 lf2 -> Int_SF lf1 l1 = Int_SF lf2 l2.forall (f : R -> R) (l1 l2 lf1 lf2 : Rlist) (a b : R),
adapted_couple f a b l1 lf1 ->
adapted_couple_opt f a b l2 lf2 ->
Int_SF lf1 l1 = Int_SF lf2 l2
Proof.forall (f : R -> R) (l1 l2 lf1 lf2 : Rlist) (a b : R),
adapted_couple f a b l1 lf1 ->
adapted_couple_opt f a b l2 lf2 ->
Int_SF lf1 l1 = Int_SF lf2 l2
  intros; destruct (Rle_dec a b) as [Hle|Hnle];
    [ apply (StepFun_P14 Hle H H0)
      | assert (H1 : b <= a);
        [ auto with real
          | eapply StepFun_P14;
            [ apply H1 | apply StepFun_P2; apply H | apply StepFun_P12; apply H0 ] ] ].
Qed.

Lemma StepFun_P16 :
  forall (f:R -> R) (l lf:Rlist) (a b:R),
    adapted_couple f a b l lf ->
    exists l' : Rlist,
      (exists lf' : Rlist, adapted_couple_opt f a b l' lf').forall (f : R -> R) (l lf : Rlist) (a b : R),
adapted_couple f a b l lf ->
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
Proof.forall (f : R -> R) (l lf : Rlist) (a b : R),
adapted_couple f a b l lf ->
exists l' lf' : Rlist, adapted_couple_opt f a b l' lf'
  intros; destruct (Rle_dec a b) as [Hle|Hnle];
    [ apply (StepFun_P10 Hle H)
      | assert (H1 : b <= a);
        [ auto with real
          | assert (H2 := StepFun_P10 H1 (StepFun_P2 H)); elim H2;
            intros l' [lf' H3]; exists l'; exists lf'; apply StepFun_P12;
              assumption ] ].
Qed.

Lemma StepFun_P17 :
  forall (f:R -> R) (l1 l2 lf1 lf2:Rlist) (a b:R),
    adapted_couple f a b l1 lf1 ->
    adapted_couple f a b l2 lf2 -> Int_SF lf1 l1 = Int_SF lf2 l2.forall (f : R -> R) (l1 l2 lf1 lf2 : Rlist) (a b : R),
adapted_couple f a b l1 lf1 ->
adapted_couple f a b l2 lf2 ->
Int_SF lf1 l1 = Int_SF lf2 l2
Proof.forall (f : R -> R) (l1 l2 lf1 lf2 : Rlist) (a b : R),
adapted_couple f a b l1 lf1 ->
adapted_couple f a b l2 lf2 ->
Int_SF lf1 l1 = Int_SF lf2 l2
  intros; elim (StepFun_P16 H); intros l' [lf' H1]; rewrite (StepFun_P15 H H1);
    rewrite (StepFun_P15 H0 H1); reflexivity.
Qed.

Lemma StepFun_P18 :
  forall a b c:R, RiemannInt_SF (mkStepFun (StepFun_P4 a b c)) = c * (b - a).forall a b c : R,
RiemannInt_SF
  {| fe := fct_cte c; pre := StepFun_P4 a b c |} =
c * (b - a)
Proof.forall a b c : R,
RiemannInt_SF
  {| fe := fct_cte c; pre := StepFun_P4 a b c |} =
c * (b - a)
  intros; unfold RiemannInt_SF; case (Rle_dec a b); intro.a, b, c:R
r:a <= b
Int_SF
  (subdivision_val
     {| fe := fct_cte c; pre := StepFun_P4 a b c |})
  (subdivision
     {| fe := fct_cte c; pre := StepFun_P4 a b c |}) =
c * (b - a)
a, b, c:R
n:~ a <= b
-
Int_SF
  (subdivision_val
     {| fe := fct_cte c; pre := StepFun_P4 a b c |})
  (subdivision
     {| fe := fct_cte c; pre := StepFun_P4 a b c |}) =
c * (b - a)
  replace
  (Int_SF (subdivision_val (mkStepFun (StepFun_P4 a b c)))
    (subdivision (mkStepFun (StepFun_P4 a b c)))) with
  (Int_SF (cons c nil) (cons a (cons b nil)));
  [ simpl; ring
    | apply StepFun_P17 with (fct_cte c) a b;
      [ apply StepFun_P3; assumption
        | apply (StepFun_P1 (mkStepFun (StepFun_P4 a b c))) ] ].a, b, c:R
n:~ a <= b
-
Int_SF
  (subdivision_val
     {| fe := fct_cte c; pre := StepFun_P4 a b c |})
  (subdivision
     {| fe := fct_cte c; pre := StepFun_P4 a b c |}) =
c * (b - a)
  replace
  (Int_SF (subdivision_val (mkStepFun (StepFun_P4 a b c)))
    (subdivision (mkStepFun (StepFun_P4 a b c)))) with
  (Int_SF (cons c nil) (cons b (cons a nil)));
  [ simpl; ring
    | apply StepFun_P17 with (fct_cte c) a b;
      [ apply StepFun_P2; apply StepFun_P3; auto with real
        | apply (StepFun_P1 (mkStepFun (StepFun_P4 a b c))) ] ].
Qed.

Lemma StepFun_P19 :
  forall (l1:Rlist) (f g:R -> R) (l:R),
    Int_SF (FF l1 (fun x:R => f x + l * g x)) l1 =
    Int_SF (FF l1 f) l1 + l * Int_SF (FF l1 g) l1.forall (l1 : Rlist) (f g : R -> R) (l : R),
Int_SF (FF l1 (fun x : R => f x + l * g x)) l1 =
Int_SF (FF l1 f) l1 + l * Int_SF (FF l1 g) l1
Proof.forall (l1 : Rlist) (f g : R -> R) (l : R),
Int_SF (FF l1 (fun x : R => f x + l * g x)) l1 =
Int_SF (FF l1 f) l1 + l * Int_SF (FF l1 g) l1
  intros; induction  l1 as [| r l1 Hrecl1];
    [ simpl; ring
      | induction  l1 as [| r0 l1 Hrecl0]; simpl;
        [ ring | simpl in Hrecl1; rewrite Hrecl1; ring ] ].
Qed.

Lemma StepFun_P20 :
  forall (l:Rlist) (f:R -> R),
    (0 < Rlength l)%nat -> Rlength l = S (Rlength (FF l f)).forall (l : Rlist) (f : R -> R),
(0 < Rlength l)%nat ->
Rlength l = S (Rlength (FF l f))
Proof.forall (l : Rlist) (f : R -> R),
(0 < Rlength l)%nat ->
Rlength l = S (Rlength (FF l f))
  intros l f H; induction l;
    [ elim (lt_irrefl _ H)
      | simpl; rewrite RList_P18; rewrite RList_P14; reflexivity ].
Qed.

Lemma StepFun_P21 :
  forall (a b:R) (f:R -> R) (l:Rlist),
    is_subdivision f a b l -> adapted_couple f a b l (FF l f).forall (a b : R) (f : R -> R) (l : Rlist),
is_subdivision f a b l ->
adapted_couple f a b l (FF l f)
Proof.forall (a b : R) (f : R -> R) (l : Rlist),
is_subdivision f a b l ->
adapted_couple f a b l (FF l f)
  intros * (x & H & H1 & H0 & H2 & H4).a, b:R
f:R -> R
l, x:Rlist
H:ordered_Rlist l
H1:pos_Rl l 0 = Rmin a b
H0:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H2:Rlength l = S (Rlength x)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl x i)
adapted_couple f a b l (FF l f)
  repeat split; try assumption.a, b:R
f:R -> R
l, x:Rlist
H:ordered_Rlist l
H1:pos_Rl l 0 = Rmin a b
H0:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H2:Rlength l = S (Rlength x)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl x i)
Rlength l = S (Rlength (FF l f))
a, b:R
f:R -> R
l, x:Rlist
H:ordered_Rlist l
H1:pos_Rl l 0 = Rmin a b
H0:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H2:Rlength l = S (Rlength x)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl x i)
forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl (FF l f) i)
  apply StepFun_P20; rewrite H2; apply lt_O_Sn.a, b:R
f:R -> R
l, x:Rlist
H:ordered_Rlist l
H1:pos_Rl l 0 = Rmin a b
H0:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H2:Rlength l = S (Rlength x)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl x i)
forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl (FF l f) i)
  intros; assert (H5 := H4 _ H3); unfold constant_D_eq, open_interval in H5;
    unfold constant_D_eq, open_interval; intros;
      induction  l as [| r l Hrecl].a, b:R
f:R -> R
x:Rlist
H:ordered_Rlist nil
H1:pos_Rl nil 0 = Rmin a b
H0:pos_Rl nil (Init.Nat.pred (Rlength nil)) =
Rmax a b
H2:Rlength nil = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq f
  (open_interval (pos_Rl nil i0)
     (pos_Rl nil (S i0))) (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength nil))%nat
H5:forall x1 : R,
pos_Rl nil i < x1 < pos_Rl nil (S i) ->
f x1 = pos_Rl x i
x0:R
H6:pos_Rl nil i < x0 < pos_Rl nil (S i)
f x0 = pos_Rl (FF nil f) i
a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
f x0 = pos_Rl (FF (cons r l) f) i
  discriminate.a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
f x0 = pos_Rl (FF (cons r l) f) i
  unfold FF; rewrite RList_P12.a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
f x0 = f (pos_Rl (mid_Rlist l r) i)
a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
(i < Rlength (mid_Rlist l r))%nat
  simpl;
    change (f x0 = f (pos_Rl (mid_Rlist (cons r l) r) (S i)));
      rewrite RList_P13; try assumption; rewrite (H5 x0 H6);
        rewrite H5.a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
pos_Rl x i = pos_Rl x i
a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
pos_Rl (cons r l) i <
(pos_Rl (cons r l) i + pos_Rl (cons r l) (S i)) / 2 <
pos_Rl (cons r l) (S i)
a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
(i < Rlength (mid_Rlist l r))%nat
  reflexivity.a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
pos_Rl (cons r l) i <
(pos_Rl (cons r l) i + pos_Rl (cons r l) (S i)) / 2 <
pos_Rl (cons r l) (S i)
a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
(i < Rlength (mid_Rlist l r))%nat
  split.a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
pos_Rl (cons r l) i <
(pos_Rl (cons r l) i + pos_Rl (cons r l) (S i)) / 2
a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
(pos_Rl (cons r l) i + pos_Rl (cons r l) (S i)) / 2 <
pos_Rl (cons r l) (S i)
a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
(i < Rlength (mid_Rlist l r))%nat
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; elim H6;
            intros; apply Rlt_trans with x0; assumption
            | discrR ] ].a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
(pos_Rl (cons r l) i + pos_Rl (cons r l) (S i)) / 2 <
pos_Rl (cons r l) (S i)
a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
(i < Rlength (mid_Rlist l r))%nat
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double;
            rewrite (Rplus_comm (pos_Rl (cons r l) i));
              apply Rplus_lt_compat_l; elim H6; intros; apply Rlt_trans with x0;
                assumption
            | discrR ] ].a, b:R
f:R -> R
r:R
l, x:Rlist
H:ordered_Rlist (cons r l)
H1:pos_Rl (cons r l) 0 = Rmin a b
H0:pos_Rl (cons r l)
  (Init.Nat.pred (Rlength (cons r l))) = 
Rmax a b
H2:Rlength (cons r l) = S (Rlength x)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l) i0)
     (pos_Rl (cons r l) (S i0))) 
  (pos_Rl x i0)
i:nat
H3:(i < Init.Nat.pred (Rlength (cons r l)))%nat
H5:forall x1 : R,
pos_Rl (cons r l) i < x1 <
pos_Rl (cons r l) (S i) -> 
f x1 = pos_Rl x i
x0:R
H6:pos_Rl (cons r l) i < x0 <
pos_Rl (cons r l) (S i)
Hrecl:ordered_Rlist l ->
pos_Rl l 0 = Rmin a b ->
pos_Rl l (Init.Nat.pred (Rlength l)) =
Rmax a b ->
Rlength l = S (Rlength x) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l i0)
      (pos_Rl l (S i0))) 
   (pos_Rl x i0)) ->
(i < Init.Nat.pred (Rlength l))%nat ->
(forall x1 : R,
 pos_Rl l i < x1 < pos_Rl l (S i) ->
 f x1 = pos_Rl x i) ->
pos_Rl l i < x0 < pos_Rl l (S i) ->
f x0 = pos_Rl (FF l f) i
(i < Rlength (mid_Rlist l r))%nat
  rewrite RList_P14; simpl in H3; apply H3.
Qed.

Lemma StepFun_P22 :
  forall (a b:R) (f g:R -> R) (lf lg:Rlist),
    a <= b ->
    is_subdivision f a b lf ->
    is_subdivision g a b lg -> is_subdivision f a b (cons_ORlist lf lg).forall (a b : R) (f g : R -> R) (lf lg : Rlist),
a <= b ->
is_subdivision f a b lf ->
is_subdivision g a b lg ->
is_subdivision f a b (cons_ORlist lf lg)
Proof.forall (a b : R) (f g : R -> R) (lf lg : Rlist),
a <= b ->
is_subdivision f a b lf ->
is_subdivision g a b lg ->
is_subdivision f a b (cons_ORlist lf lg)
  unfold is_subdivision; intros a b f g lf lg Hyp X X0; elim X; elim X0;
    clear X X0; intros lg0 p lf0 p0; assert (Hyp_min : Rmin a b = a).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0:Rlist
p:adapted_couple g a b lg lg0
lf0:Rlist
p0:adapted_couple f a b lf lf0
Rmin a b = a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0:Rlist
p:adapted_couple g a b lg lg0
lf0:Rlist
p0:adapted_couple f a b lf lf0
Hyp_min:Rmin a b = a
{l0 : Rlist &
adapted_couple f a b (cons_ORlist lf lg) l0}
  unfold Rmin; decide (Rle_dec a b) with Hyp; reflexivity.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0:Rlist
p:adapted_couple g a b lg lg0
lf0:Rlist
p0:adapted_couple f a b lf lf0
Hyp_min:Rmin a b = a
{l0 : Rlist &
adapted_couple f a b (cons_ORlist lf lg) l0}
  assert (Hyp_max : Rmax a b = b).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0:Rlist
p:adapted_couple g a b lg lg0
lf0:Rlist
p0:adapted_couple f a b lf lf0
Hyp_min:Rmin a b = a
Rmax a b = b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0:Rlist
p:adapted_couple g a b lg lg0
lf0:Rlist
p0:adapted_couple f a b lf lf0
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
{l0 : Rlist &
adapted_couple f a b (cons_ORlist lf lg) l0}
  unfold Rmax; decide (Rle_dec a b) with Hyp; reflexivity.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0:Rlist
p:adapted_couple g a b lg lg0
lf0:Rlist
p0:adapted_couple f a b lf lf0
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
{l0 : Rlist &
adapted_couple f a b (cons_ORlist lf lg) l0}
  apply existT with (FF (cons_ORlist lf lg) f); unfold adapted_couple in p, p0;
    decompose [and] p; decompose [and] p0; clear p p0;
      rewrite Hyp_min in H6; rewrite Hyp_min in H1; rewrite Hyp_max in H0;
        rewrite Hyp_max in H5; unfold adapted_couple;
          repeat split.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
ordered_Rlist (cons_ORlist lf lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 = Rmin a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  apply RList_P2; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 = Rmin a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  rewrite Hyp_min; symmetry ; apply Rle_antisym.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
a <= pos_Rl (cons_ORlist lf lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  induction  lf as [| r lf Hreclf].a, b:R
f, g:R -> R
lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist nil
H6:pos_Rl nil 0 = a
H5:pos_Rl nil (Init.Nat.pred (Rlength nil)) = b
H7:Rlength nil = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq f
  (open_interval (pos_Rl nil i)
     (pos_Rl nil (S i))) 
  (pos_Rl lf0 i)
a <= pos_Rl (cons_ORlist nil lg) 0
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  simpl; right; symmetry ; assumption.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  assert
    (H10 :
      In (pos_Rl (cons_ORlist (cons r lf) lg) 0) (cons_ORlist (cons r lf) lg)).a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg)
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
H10:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg)
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  elim
    (RList_P3 (cons_ORlist (cons r lf) lg)
      (pos_Rl (cons_ORlist (cons r lf) lg) 0)); intros _ H10;
    apply H10; exists 0%nat; split;
      [ reflexivity | rewrite RList_P11; simpl; apply lt_O_Sn ].a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
H10:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg)
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) 0));
    intros H12 _; assert (H13 := H12 H10); elim H13; intro.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
H10:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg)
H12:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg) ->
In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf) \/
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
H13:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf) \/
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
H8:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf)
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
H10:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg)
H12:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg) ->
In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf) \/
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
H13:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf) \/
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
H8:In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) 0));
    intros H11 _; assert (H14 := H11 H8); elim H14; intros;
      elim H15; clear H15; intros; rewrite H15; rewrite <- H6;
        elim (RList_P6 (cons r lf)); intros; apply H17;
          [ assumption | apply le_O_n | assumption ].a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
H10:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg)
H12:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg) ->
In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf) \/
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
H13:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf) \/
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
H8:In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) 0)); intros H11 _;
    assert (H14 := H11 H8); elim H14; intros; elim H15;
      clear H15; intros; rewrite H15; rewrite <- H1; elim (RList_P6 lg);
        intros; apply H17; [ assumption | apply le_O_n | assumption ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  induction  lf as [| r lf Hreclf].a, b:R
f, g:R -> R
lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist nil
H6:pos_Rl nil 0 = a
H5:pos_Rl nil (Init.Nat.pred (Rlength nil)) = b
H7:Rlength nil = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq f
  (open_interval (pos_Rl nil i)
     (pos_Rl nil (S i))) 
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist nil lg) 0 <= a
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg) 0 <= a
pos_Rl (cons_ORlist (cons r lf) lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  simpl; right; assumption.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg) 0 <= a
pos_Rl (cons_ORlist (cons r lf) lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  assert (H8 : In a (cons_ORlist (cons r lf) lg)).a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg) 0 <= a
In a (cons_ORlist (cons r lf) lg)
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg) 0 <= a
H8:In a (cons_ORlist (cons r lf) lg)
pos_Rl (cons_ORlist (cons r lf) lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  elim (RList_P9 (cons r lf) lg a); intros; apply H10; left;
    elim (RList_P3 (cons r lf) a); intros; apply H12;
      exists 0%nat; split;
        [ symmetry ; assumption | simpl; apply lt_O_Sn ].a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg) 0 <= a
H8:In a (cons_ORlist (cons r lf) lg)
pos_Rl (cons_ORlist (cons r lf) lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  apply RList_P5; [ apply RList_P2; assumption | assumption ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  rewrite Hyp_max; apply Rle_antisym.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  induction  lf as [| r lf Hreclf].a, b:R
f, g:R -> R
lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist nil
H6:pos_Rl nil 0 = a
H5:pos_Rl nil (Init.Nat.pred (Rlength nil)) = b
H7:Rlength nil = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq f
  (open_interval (pos_Rl nil i)
     (pos_Rl nil (S i))) 
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist nil lg)
  (Init.Nat.pred (Rlength (cons_ORlist nil lg))) <= b
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  simpl; right; assumption.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  assert
    (H8 :
      In
      (pos_Rl (cons_ORlist (cons r lf) lg)
        (pred (Rlength (cons_ORlist (cons r lf) lg))))
      (cons_ORlist (cons r lf) lg)).a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  elim
    (RList_P3 (cons_ORlist (cons r lf) lg)
      (pos_Rl (cons_ORlist (cons r lf) lg)
        (pred (Rlength (cons_ORlist (cons r lf) lg)))));
    intros _ H10; apply H10;
      exists (pred (Rlength (cons_ORlist (cons r lf) lg)));
        split; [ reflexivity | rewrite RList_P11; simpl; apply lt_n_Sn ].a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  elim
    (RList_P9 (cons r lf) lg
      (pos_Rl (cons_ORlist (cons r lf) lg)
        (pred (Rlength (cons_ORlist (cons r lf) lg)))));
    intros H10 _.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  assert (H11 := H10 H8); elim H11; intro.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H11:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H12:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf)
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H11:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H12:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  elim
    (RList_P3 (cons r lf)
      (pos_Rl (cons_ORlist (cons r lf) lg)
        (pred (Rlength (cons_ORlist (cons r lf) lg)))));
    intros H13 _; assert (H14 := H13 H12); elim H14; intros;
      elim H15; clear H15; intros; rewrite H15; rewrite <- H5;
        elim (RList_P6 (cons r lf)); intros; apply H17;
          [ assumption
            | simpl; simpl in H14; apply lt_n_Sm_le; assumption
            | simpl; apply lt_n_Sn ].a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H11:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H12:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  elim
    (RList_P3 lg
      (pos_Rl (cons_ORlist (cons r lf) lg)
        (pred (Rlength (cons_ORlist (cons r lf) lg)))));
    intros H13 _; assert (H14 := H13 H12); elim H14; intros;
      elim H15; clear H15; intros.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H11:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H12:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H13:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg ->
exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
H14:exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
x:nat
H15:pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) =
pos_Rl lg x
H16:(x < Rlength lg)%nat
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  rewrite H15; assert (H17 : Rlength lg = S (pred (Rlength lg))).a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H11:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H12:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H13:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg ->
exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
H14:exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
x:nat
H15:pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) =
pos_Rl lg x
H16:(x < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H11:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H12:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H13:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg ->
exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
H14:exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
x:nat
H15:pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) =
pos_Rl lg x
H16:(x < Rlength lg)%nat
H17:Rlength lg = S (Init.Nat.pred (Rlength lg))
pos_Rl lg x <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  apply S_pred with 0%nat; apply neq_O_lt; red; intro;
    rewrite <- H17 in H16; elim (lt_n_O _ H16).a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H11:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H12:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H13:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg ->
exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
H14:exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
x:nat
H15:pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) =
pos_Rl lg x
H16:(x < Rlength lg)%nat
H17:Rlength lg = S (Init.Nat.pred (Rlength lg))
pos_Rl lg x <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  rewrite <- H0; elim (RList_P6 lg); intros; apply H18;
    [ assumption
      | rewrite H17 in H16; apply lt_n_Sm_le; assumption
      | apply lt_pred_n_n; rewrite H17; apply lt_O_Sn ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  induction  lf as [| r lf Hreclf].a, b:R
f, g:R -> R
lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist nil
H6:pos_Rl nil 0 = a
H5:pos_Rl nil (Init.Nat.pred (Rlength nil)) = b
H7:Rlength nil = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq f
  (open_interval (pos_Rl nil i)
     (pos_Rl nil (S i))) 
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist nil lg)
  (Init.Nat.pred (Rlength (cons_ORlist nil lg)))
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg)))
b <=
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  simpl; right; symmetry ; assumption.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg)))
b <=
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  assert (H8 : In b (cons_ORlist (cons r lf) lg)).a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg)))
In b (cons_ORlist (cons r lf) lg)
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg)))
H8:In b (cons_ORlist (cons r lf) lg)
b <=
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  elim (RList_P9 (cons r lf) lg b); intros; apply H10; left;
    elim (RList_P3 (cons r lf) b); intros; apply H12;
      exists (pred (Rlength (cons r lf))); split;
        [ symmetry ; assumption | simpl; apply lt_n_Sn ].a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg)))
H8:In b (cons_ORlist (cons r lf) lg)
b <=
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  apply RList_P7; [ apply RList_P2; assumption | assumption ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) f))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  apply StepFun_P20; rewrite RList_P11; rewrite H2; rewrite H7; simpl;
    apply lt_O_Sn.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) f) i)
  intros; unfold constant_D_eq, open_interval; intros;
    cut
      (exists l : R,
        constant_D_eq f
        (open_interval (pos_Rl (cons_ORlist lf lg) i)
          (pos_Rl (cons_ORlist lf lg) (S i))) l).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
(exists l : R,
   constant_D_eq f
     (open_interval (pos_Rl (cons_ORlist lf lg) i)
        (pos_Rl (cons_ORlist lf lg) (S i))) l) ->
f x = pos_Rl (FF (cons_ORlist lf lg) f) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  intros; elim H11; clear H11; intros; assert (H12 := H11);
    assert
      (Hyp_cons :
        exists r : R, (exists r0 : Rlist, cons_ORlist lf lg = cons r r0)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
exists (r : R) (r0 : Rlist),
  cons_ORlist lf lg = cons r r0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
Hyp_cons:exists (r : R) (r0 : Rlist),
  cons_ORlist lf lg = cons r r0
f x = pos_Rl (FF (cons_ORlist lf lg) f) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply RList_P19; red; intro; rewrite H13 in H8; elim (lt_n_O _ H8).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
Hyp_cons:exists (r : R) (r0 : Rlist),
  cons_ORlist lf lg = cons r r0
f x = pos_Rl (FF (cons_ORlist lf lg) f) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  elim Hyp_cons; clear Hyp_cons; intros r [r0 Hyp_cons]; rewrite Hyp_cons;
    unfold FF; rewrite RList_P12.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
f x = f (pos_Rl (mid_Rlist r0 r) i)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  change (f x = f (pos_Rl (mid_Rlist (cons r r0) r) (S i)));
    rewrite <- Hyp_cons; rewrite RList_P13.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
f x =
f
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  assert (H13 := RList_P2 _ _ H _ H8); elim H13; intro.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
f x =
f
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i =
pos_Rl (cons_ORlist lf lg) (S i)
f x =
f
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  unfold constant_D_eq, open_interval in H11, H12; rewrite (H11 x H10);
    assert
      (H15 :
        pos_Rl (cons_ORlist lf lg) i <
        (pos_Rl (cons_ORlist lf lg) i + pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
        pos_Rl (cons_ORlist lf lg) (S i)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
f x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i <
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
f x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
H15:pos_Rl (cons_ORlist lf lg) i <
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
x0 =
f
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i =
pos_Rl (cons_ORlist lf lg) (S i)
f x =
f
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  split.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
f x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i <
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
f x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
f x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
H15:pos_Rl (cons_ORlist lf lg) i <
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
x0 =
f
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i =
pos_Rl (cons_ORlist lf lg) (S i)
f x =
f
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
            | discrR ] ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
f x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
f x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
H15:pos_Rl (cons_ORlist lf lg) i <
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
x0 =
f
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i =
pos_Rl (cons_ORlist lf lg) (S i)
f x =
f
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double;
            rewrite (Rplus_comm (pos_Rl (cons_ORlist lf lg) i));
              apply Rplus_lt_compat_l; assumption
            | discrR ] ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
f x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
H15:pos_Rl (cons_ORlist lf lg) i <
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
x0 =
f
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i =
pos_Rl (cons_ORlist lf lg) (S i)
f x =
f
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  rewrite (H11 _ H15); reflexivity.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i =
pos_Rl (cons_ORlist lf lg) (S i)
f x =
f
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  elim H10; intros; rewrite H14 in H15;
    elim (Rlt_irrefl _ (Rlt_trans _ _ _ H16 H15)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply H8.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq f
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  rewrite RList_P14; rewrite Hyp_cons in H8; simpl in H8; apply H8.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  assert (H11 : a < b).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
a < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
a <= pos_Rl (cons_ORlist lf lg) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  rewrite <- H6; rewrite <- (RList_P15 lf lg).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) 0 <=
pos_Rl (cons_ORlist lf lg) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
ordered_Rlist lf
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
ordered_Rlist lg
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf 0 = pos_Rl lg 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
ordered_Rlist (cons_ORlist lf lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(0 <= i)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(i < Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
ordered_Rlist lf
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
ordered_Rlist lg
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf 0 = pos_Rl lg 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply RList_P2; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(0 <= i)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(i < Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
ordered_Rlist lf
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
ordered_Rlist lg
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf 0 = pos_Rl lg 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply le_O_n.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(i < Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
ordered_Rlist lf
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
ordered_Rlist lg
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf 0 = pos_Rl lg 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply lt_trans with (pred (Rlength (cons_ORlist lf lg)));
    [ assumption
      | apply lt_pred_n_n; apply neq_O_lt; red; intro;
        rewrite <- H13 in H8; elim (lt_n_O _ H8) ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
ordered_Rlist lf
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
ordered_Rlist lg
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf 0 = pos_Rl lg 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
ordered_Rlist lg
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf 0 = pos_Rl lg 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf 0 = pos_Rl lg 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  rewrite H1; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) (S i) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  elim H10; intros; apply Rlt_trans with x; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) (S i) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  rewrite <- H5; rewrite <- (RList_P16 lf lg); try assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) (S i) <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf (Init.Nat.pred (Rlength lf)) =
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
ordered_Rlist (cons_ORlist lf lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(S i <= Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf (Init.Nat.pred (Rlength lf)) =
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply RList_P2; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(S i <= Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf (Init.Nat.pred (Rlength lf)) =
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply lt_n_Sm_le; apply lt_n_S; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf (Init.Nat.pred (Rlength lf)) =
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H13 in H8;
    elim (lt_n_O _ H8).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf (Init.Nat.pred (Rlength lf)) =
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  rewrite H0; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  set
    (I :=
      fun j:nat =>
        pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\ (j < Rlength lf)%nat);
    assert (H12 : Nbound I).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
Nbound I
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  unfold Nbound; exists (Rlength lf); intros; unfold I in H12; elim H12;
    intros; apply lt_le_weak; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  assert (H13 :  exists n : nat, I n).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
exists n : nat, I n
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  exists 0%nat; unfold I; split.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
pos_Rl lf 0 <= pos_Rl (cons_ORlist lf lg) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lf)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply Rle_trans with (pos_Rl (cons_ORlist lf lg) 0).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
pos_Rl lf 0 <= pos_Rl (cons_ORlist lf lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
pos_Rl (cons_ORlist lf lg) 0 <=
pos_Rl (cons_ORlist lf lg) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lf)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  right; symmetry .a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
pos_Rl (cons_ORlist lf lg) 0 = pos_Rl lf 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
pos_Rl (cons_ORlist lf lg) 0 <=
pos_Rl (cons_ORlist lf lg) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lf)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply RList_P15; try assumption; rewrite H1; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
pos_Rl (cons_ORlist lf lg) 0 <=
pos_Rl (cons_ORlist lf lg) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lf)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  elim (RList_P6 (cons_ORlist lf lg)); intros; apply H13.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H14:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
ordered_Rlist (cons_ORlist lf lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H14:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(0 <= i)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H14:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(i < Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lf)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply RList_P2; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H14:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(0 <= i)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H14:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(i < Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lf)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply le_O_n.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H14:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(i < Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lf)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply lt_trans with (pred (Rlength (cons_ORlist lf lg))).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H14:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H14:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lf)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H14:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lf)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H15 in H8;
    elim (lt_n_O _ H8).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lf)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply neq_O_lt; red; intro; rewrite <- H13 in H5;
    rewrite <- H6 in H11; rewrite <- H5 in H11; elim (Rlt_irrefl _ H11).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq f
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  assert (H14 := Nzorn H13 H12); elim H14; clear H14; intros x0 H14;
    exists (pos_Rl lf0 x0); unfold constant_D_eq, open_interval;
      intros; assert (H16 := H9 x0); assert (H17 : (x0 < pred (Rlength lf))%nat).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
(x0 < Init.Nat.pred (Rlength lf))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  elim H14; clear H14; intros; unfold I in H14; elim H14; clear H14; intros;
    apply lt_S_n; replace (S (pred (Rlength lf))) with (Rlength lf).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
(S x0 < Rlength lf)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  inversion H18.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
(S x0 < S x0)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
m:nat
H20:(S x0 <= m)%nat
H19:S m = Rlength lf
(S x0 < S m)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  2: apply lt_n_S; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
(S x0 < S x0)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  cut (x0 = pred (Rlength lf)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
x0 = Init.Nat.pred (Rlength lf) -> (S x0 < S x0)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
x0 = Init.Nat.pred (Rlength lf)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  intro; rewrite H19 in H14; rewrite H5 in H14;
    cut (pos_Rl (cons_ORlist lf lg) i < b).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
pos_Rl (cons_ORlist lf lg) i < b -> (S x0 < S x0)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
x0 = Init.Nat.pred (Rlength lf)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H14 H21)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
x0 = Init.Nat.pred (Rlength lf)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
pos_Rl (cons_ORlist lf lg) (S i) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
x0 = Init.Nat.pred (Rlength lf)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  elim H10; intros; apply Rlt_trans with x; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
pos_Rl (cons_ORlist lf lg) (S i) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
x0 = Init.Nat.pred (Rlength lf)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  rewrite <- H5;
    apply Rle_trans with
      (pos_Rl (cons_ORlist lf lg) (pred (Rlength (cons_ORlist lf lg)))).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
pos_Rl (cons_ORlist lf lg) (S i) <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) <=
pos_Rl lf (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
x0 = Init.Nat.pred (Rlength lf)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  elim (RList_P6 (cons_ORlist lf lg)); intros; apply H21.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
H21:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H22:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
ordered_Rlist (cons_ORlist lf lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
H21:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H22:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(S i <= Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
H21:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H22:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) <=
pos_Rl lf (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
x0 = Init.Nat.pred (Rlength lf)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  apply RList_P2; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
H21:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H22:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(S i <= Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
H21:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H22:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) <=
pos_Rl lf (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
x0 = Init.Nat.pred (Rlength lf)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  apply lt_n_Sm_le; apply lt_n_S; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
H21:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H22:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) <=
pos_Rl lf (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
x0 = Init.Nat.pred (Rlength lf)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H23 in H8;
    elim (lt_n_O _ H8).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
H19:x0 = Init.Nat.pred (Rlength lf)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) <=
pos_Rl lf (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
x0 = Init.Nat.pred (Rlength lf)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  right; apply RList_P16; try assumption; rewrite H0; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
H20:S x0 = Rlength lf
x0 = Init.Nat.pred (Rlength lf)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  rewrite <- H20; reflexivity.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lf)%nat
Rlength lf = S (Init.Nat.pred (Rlength lf))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  apply S_pred with 0%nat; apply neq_O_lt; red; intro;
    rewrite <- H19 in H18; elim (lt_n_O _ H18).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
f x1 = pos_Rl lf0 x0
  assert (H18 := H16 H17); unfold constant_D_eq, open_interval in H18;
    rewrite (H18 x1).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
pos_Rl lf0 x0 = pos_Rl lf0 x0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
pos_Rl lf x0 < x1 < pos_Rl lf (S x0)
  reflexivity.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
pos_Rl lf x0 < x1 < pos_Rl lf (S x0)
  elim H15; clear H15; intros; elim H14; clear H14; intros; unfold I in H14;
    elim H14; clear H14; intros; split.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
pos_Rl lf x0 < x1
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
x1 < pos_Rl lf (S x0)
  apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i); assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
x1 < pos_Rl lf (S x0)
  apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)); try assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lf (S x0)
  assert (H22 : (S x0 < Rlength lf)%nat).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
(S x0 < Rlength lf)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lf (S x0)
  replace (Rlength lf) with (S (pred (Rlength lf)));
  [ apply lt_n_S; assumption
    | symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red;
      intro; rewrite <- H22 in H21; elim (lt_n_O _ H21) ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lf (S x0)
  elim (Rle_dec (pos_Rl lf (S x0)) (pos_Rl (cons_ORlist lf lg) i)); intro a0.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
a0:pos_Rl lf (S x0) <= pos_Rl (cons_ORlist lf lg) i
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lf (S x0)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
a0:~
pos_Rl lf (S x0) <= pos_Rl (cons_ORlist lf lg) i
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lf (S x0)
  assert (H23 : (S x0 <= x0)%nat).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
a0:pos_Rl lf (S x0) <= pos_Rl (cons_ORlist lf lg) i
(S x0 <= x0)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
a0:pos_Rl lf (S x0) <= pos_Rl (cons_ORlist lf lg) i
H23:(S x0 <= x0)%nat
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lf (S x0)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
a0:~
pos_Rl lf (S x0) <= pos_Rl (cons_ORlist lf lg) i
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lf (S x0)
  apply H20; unfold I; split; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
a0:pos_Rl lf (S x0) <= pos_Rl (cons_ORlist lf lg) i
H23:(S x0 <= x0)%nat
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lf (S x0)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
a0:~
pos_Rl lf (S x0) <= pos_Rl (cons_ORlist lf lg) i
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lf (S x0)
  elim (le_Sn_n _ H23).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
a0:~
pos_Rl lf (S x0) <= pos_Rl (cons_ORlist lf lg) i
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lf (S x0)
  assert (H23 : pos_Rl (cons_ORlist lf lg) i < pos_Rl lf (S x0)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
a0:~
pos_Rl lf (S x0) <= pos_Rl (cons_ORlist lf lg) i
pos_Rl (cons_ORlist lf lg) i < pos_Rl lf (S x0)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
a0:~
pos_Rl lf (S x0) <= pos_Rl (cons_ORlist lf lg) i
H23:pos_Rl (cons_ORlist lf lg) i < pos_Rl lf (S x0)
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lf (S x0)
  auto with real.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
a0:~
pos_Rl lf (S x0) <= pos_Rl (cons_ORlist lf lg) i
H23:pos_Rl (cons_ORlist lf lg) i < pos_Rl lf (S x0)
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lf (S x0)
  clear a0; apply RList_P17; try assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
H23:pos_Rl (cons_ORlist lf lg) i < pos_Rl lf (S x0)
ordered_Rlist (cons_ORlist lf lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
H23:pos_Rl (cons_ORlist lf lg) i < pos_Rl lf (S x0)
In (pos_Rl lf (S x0)) (cons_ORlist lf lg)
  apply RList_P2; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lf)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf x0)
     (pos_Rl lf (S x0))) 
  (pos_Rl lf0 x0)
H17:(x0 < Init.Nat.pred (Rlength lf))%nat
H18:forall x2 : R,
pos_Rl lf x0 < x2 < pos_Rl lf (S x0) ->
f x2 = pos_Rl lf0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lf x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lf)%nat
H22:(S x0 < Rlength lf)%nat
H23:pos_Rl (cons_ORlist lf lg) i < pos_Rl lf (S x0)
In (pos_Rl lf (S x0)) (cons_ORlist lf lg)
  elim (RList_P9 lf lg (pos_Rl lf (S x0))); intros; apply H25; left;
    elim (RList_P3 lf (pos_Rl lf (S x0))); intros; apply H27;
      exists (S x0); split; [ reflexivity | apply H22 ].
Qed.

Lemma StepFun_P23 :
  forall (a b:R) (f g:R -> R) (lf lg:Rlist),
    is_subdivision f a b lf ->
    is_subdivision g a b lg -> is_subdivision f a b (cons_ORlist lf lg).forall (a b : R) (f g : R -> R) (lf lg : Rlist),
is_subdivision f a b lf ->
is_subdivision g a b lg ->
is_subdivision f a b (cons_ORlist lf lg)
Proof.forall (a b : R) (f g : R -> R) (lf lg : Rlist),
is_subdivision f a b lf ->
is_subdivision g a b lg ->
is_subdivision f a b (cons_ORlist lf lg)
  intros; case (Rle_dec a b); intro;
    [ apply StepFun_P22 with g; assumption
      | apply StepFun_P5; apply StepFun_P22 with g;
        [ auto with real
          | apply StepFun_P5; assumption
          | apply StepFun_P5; assumption ] ].
Qed.

Lemma StepFun_P24 :
  forall (a b:R) (f g:R -> R) (lf lg:Rlist),
    a <= b ->
    is_subdivision f a b lf ->
    is_subdivision g a b lg -> is_subdivision g a b (cons_ORlist lf lg).forall (a b : R) (f g : R -> R) (lf lg : Rlist),
a <= b ->
is_subdivision f a b lf ->
is_subdivision g a b lg ->
is_subdivision g a b (cons_ORlist lf lg)
Proof.forall (a b : R) (f g : R -> R) (lf lg : Rlist),
a <= b ->
is_subdivision f a b lf ->
is_subdivision g a b lg ->
is_subdivision g a b (cons_ORlist lf lg)
  unfold is_subdivision; intros a b f g lf lg Hyp X X0; elim X; elim X0;
    clear X X0; intros lg0 p lf0 p0; assert (Hyp_min : Rmin a b = a).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0:Rlist
p:adapted_couple g a b lg lg0
lf0:Rlist
p0:adapted_couple f a b lf lf0
Rmin a b = a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0:Rlist
p:adapted_couple g a b lg lg0
lf0:Rlist
p0:adapted_couple f a b lf lf0
Hyp_min:Rmin a b = a
{l0 : Rlist &
adapted_couple g a b (cons_ORlist lf lg) l0}
  unfold Rmin; decide (Rle_dec a b) with Hyp; reflexivity.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0:Rlist
p:adapted_couple g a b lg lg0
lf0:Rlist
p0:adapted_couple f a b lf lf0
Hyp_min:Rmin a b = a
{l0 : Rlist &
adapted_couple g a b (cons_ORlist lf lg) l0}
  assert (Hyp_max : Rmax a b = b).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0:Rlist
p:adapted_couple g a b lg lg0
lf0:Rlist
p0:adapted_couple f a b lf lf0
Hyp_min:Rmin a b = a
Rmax a b = b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0:Rlist
p:adapted_couple g a b lg lg0
lf0:Rlist
p0:adapted_couple f a b lf lf0
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
{l0 : Rlist &
adapted_couple g a b (cons_ORlist lf lg) l0}
  unfold Rmax; decide (Rle_dec a b) with Hyp; reflexivity.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0:Rlist
p:adapted_couple g a b lg lg0
lf0:Rlist
p0:adapted_couple f a b lf lf0
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
{l0 : Rlist &
adapted_couple g a b (cons_ORlist lf lg) l0}
  apply existT with (FF (cons_ORlist lf lg) g); unfold adapted_couple in p, p0;
    decompose [and] p; decompose [and] p0; clear p p0;
      rewrite Hyp_min in H1; rewrite Hyp_min in H6; rewrite Hyp_max in H0;
        rewrite Hyp_max in H5; unfold adapted_couple;
          repeat split.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
ordered_Rlist (cons_ORlist lf lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 = Rmin a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  apply RList_P2; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 = Rmin a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  rewrite Hyp_min; symmetry ; apply Rle_antisym.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
a <= pos_Rl (cons_ORlist lf lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  induction  lf as [| r lf Hreclf].a, b:R
f, g:R -> R
lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist nil
H6:pos_Rl nil 0 = a
H5:pos_Rl nil (Init.Nat.pred (Rlength nil)) = b
H7:Rlength nil = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq f
  (open_interval (pos_Rl nil i)
     (pos_Rl nil (S i))) 
  (pos_Rl lf0 i)
a <= pos_Rl (cons_ORlist nil lg) 0
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  simpl; right; symmetry ; assumption.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  assert
    (H10 :
      In (pos_Rl (cons_ORlist (cons r lf) lg) 0) (cons_ORlist (cons r lf) lg)).a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg)
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
H10:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg)
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  elim
    (RList_P3 (cons_ORlist (cons r lf) lg)
      (pos_Rl (cons_ORlist (cons r lf) lg) 0)); intros _ H10;
    apply H10; exists 0%nat; split;
      [ reflexivity | rewrite RList_P11; simpl; apply lt_O_Sn ].a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
H10:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg)
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) 0));
    intros H12 _; assert (H13 := H12 H10); elim H13; intro.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
H10:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg)
H12:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg) ->
In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf) \/
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
H13:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf) \/
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
H8:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf)
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
H10:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg)
H12:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg) ->
In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf) \/
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
H13:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf) \/
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
H8:In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) 0));
    intros H11 _; assert (H14 := H11 H8); elim H14; intros;
      elim H15; clear H15; intros; rewrite H15; rewrite <- H6;
        elim (RList_P6 (cons r lf)); intros; apply H17;
          [ assumption | apply le_O_n | assumption ].a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
a <= pos_Rl (cons_ORlist lf lg) 0
H10:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg)
H12:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons_ORlist (cons r lf) lg) ->
In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf) \/
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
H13:In (pos_Rl (cons_ORlist (cons r lf) lg) 0)
  (cons r lf) \/
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
H8:In (pos_Rl (cons_ORlist (cons r lf) lg) 0) lg
a <= pos_Rl (cons_ORlist (cons r lf) lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) 0)); intros H11 _;
    assert (H14 := H11 H8); elim H14; intros; elim H15;
      clear H15; intros; rewrite H15; rewrite <- H1; elim (RList_P6 lg);
        intros; apply H17; [ assumption | apply le_O_n | assumption ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  induction  lf as [| r lf Hreclf].a, b:R
f, g:R -> R
lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist nil
H6:pos_Rl nil 0 = a
H5:pos_Rl nil (Init.Nat.pred (Rlength nil)) = b
H7:Rlength nil = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq f
  (open_interval (pos_Rl nil i)
     (pos_Rl nil (S i))) 
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist nil lg) 0 <= a
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg) 0 <= a
pos_Rl (cons_ORlist (cons r lf) lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  simpl; right; assumption.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg) 0 <= a
pos_Rl (cons_ORlist (cons r lf) lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  assert (H8 : In a (cons_ORlist (cons r lf) lg)).a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg) 0 <= a
In a (cons_ORlist (cons r lf) lg)
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg) 0 <= a
H8:In a (cons_ORlist (cons r lf) lg)
pos_Rl (cons_ORlist (cons r lf) lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  elim (RList_P9 (cons r lf) lg a); intros; apply H10; left;
    elim (RList_P3 (cons r lf) a); intros; apply H12;
      exists 0%nat; split;
        [ symmetry ; assumption | simpl; apply lt_O_Sn ].a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg) 0 <= a
H8:In a (cons_ORlist (cons r lf) lg)
pos_Rl (cons_ORlist (cons r lf) lg) 0 <= a
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  apply RList_P5; [ apply RList_P2; assumption | assumption ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) =
Rmax a b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  rewrite Hyp_max; apply Rle_antisym.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  induction  lf as [| r lf Hreclf].a, b:R
f, g:R -> R
lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist nil
H6:pos_Rl nil 0 = a
H5:pos_Rl nil (Init.Nat.pred (Rlength nil)) = b
H7:Rlength nil = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq f
  (open_interval (pos_Rl nil i)
     (pos_Rl nil (S i))) 
  (pos_Rl lf0 i)
pos_Rl (cons_ORlist nil lg)
  (Init.Nat.pred (Rlength (cons_ORlist nil lg))) <= b
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  simpl; right; assumption.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  assert
    (H8 :
      In
      (pos_Rl (cons_ORlist (cons r lf) lg)
        (pred (Rlength (cons_ORlist (cons r lf) lg))))
      (cons_ORlist (cons r lf) lg)).a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  elim
    (RList_P3 (cons_ORlist (cons r lf) lg)
      (pos_Rl (cons_ORlist (cons r lf) lg)
        (pred (Rlength (cons_ORlist (cons r lf) lg)))));
    intros _ H10; apply H10;
      exists (pred (Rlength (cons_ORlist (cons r lf) lg)));
        split; [ reflexivity | rewrite RList_P11; simpl; apply lt_n_Sn ].a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  elim
    (RList_P9 (cons r lf) lg
      (pos_Rl (cons_ORlist (cons r lf) lg)
        (pred (Rlength (cons_ORlist (cons r lf) lg)))));
    intros H10 _; assert (H11 := H10 H8); elim H11; intro.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H11:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H12:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf)
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H11:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H12:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  elim
    (RList_P3 (cons r lf)
      (pos_Rl (cons_ORlist (cons r lf) lg)
        (pred (Rlength (cons_ORlist (cons r lf) lg)))));
    intros H13 _; assert (H14 := H13 H12); elim H14; intros;
      elim H15; clear H15; intros; rewrite H15; rewrite <- H5;
        elim (RList_P6 (cons r lf)); intros; apply H17;
          [ assumption
            | simpl; simpl in H14; apply lt_n_Sm_le; assumption
            | simpl; apply lt_n_Sn ].a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H11:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H12:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  elim
    (RList_P3 lg
      (pos_Rl (cons_ORlist (cons r lf) lg)
        (pred (Rlength (cons_ORlist (cons r lf) lg)))));
    intros H13 _; assert (H14 := H13 H12); elim H14; intros;
      elim H15; clear H15; intros; rewrite H15;
        assert (H17 : Rlength lg = S (pred (Rlength lg))).a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H11:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H12:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H13:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg ->
exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
H14:exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
x:nat
H15:pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) =
pos_Rl lg x
H16:(x < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H11:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H12:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H13:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg ->
exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
H14:exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
x:nat
H15:pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) =
pos_Rl lg x
H16:(x < Rlength lg)%nat
H17:Rlength lg = S (Init.Nat.pred (Rlength lg))
pos_Rl lg x <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  apply S_pred with 0%nat; apply neq_O_lt; red; intro;
    rewrite <- H17 in H16; elim (lt_n_O _ H16).a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg))) <= b
H8:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg)
H10:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons_ORlist (cons r lf) lg) ->
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H11:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  (cons r lf) \/
In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H12:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg
H13:In
  (pos_Rl (cons_ORlist (cons r lf) lg)
     (Init.Nat.pred
        (Rlength (cons_ORlist (cons r lf) lg))))
  lg ->
exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
H14:exists i : nat,
  pos_Rl (cons_ORlist (cons r lf) lg)
    (Init.Nat.pred
       (Rlength (cons_ORlist (cons r lf) lg))) =
  pos_Rl lg i /\ (i < Rlength lg)%nat
x:nat
H15:pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg))) =
pos_Rl lg x
H16:(x < Rlength lg)%nat
H17:Rlength lg = S (Init.Nat.pred (Rlength lg))
pos_Rl lg x <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  rewrite <- H0; elim (RList_P6 lg); intros; apply H18;
    [ assumption
      | rewrite H17 in H16; apply lt_n_Sm_le; assumption
      | apply lt_pred_n_n; rewrite H17; apply lt_O_Sn ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  induction  lf as [| r lf Hreclf].a, b:R
f, g:R -> R
lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist nil
H6:pos_Rl nil 0 = a
H5:pos_Rl nil (Init.Nat.pred (Rlength nil)) = b
H7:Rlength nil = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq f
  (open_interval (pos_Rl nil i)
     (pos_Rl nil (S i))) 
  (pos_Rl lf0 i)
b <=
pos_Rl (cons_ORlist nil lg)
  (Init.Nat.pred (Rlength (cons_ORlist nil lg)))
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg)))
b <=
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  simpl; right; symmetry ; assumption.a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg)))
b <=
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  assert (H8 : In b (cons_ORlist (cons r lf) lg)).a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg)))
In b (cons_ORlist (cons r lf) lg)
a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg)))
H8:In b (cons_ORlist (cons r lf) lg)
b <=
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  elim (RList_P9 (cons r lf) lg b); intros; apply H10; left;
    elim (RList_P3 (cons r lf) b); intros; apply H12;
      exists (pred (Rlength (cons r lf))); split;
        [ symmetry ; assumption | simpl; apply lt_n_Sn ].a, b:R
f, g:R -> R
r:R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist (cons r lf)
H6:pos_Rl (cons r lf) 0 = a
H5:pos_Rl (cons r lf)
  (Init.Nat.pred (Rlength (cons r lf))) = b
H7:Rlength (cons r lf) = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r lf)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r lf) i)
     (pos_Rl (cons r lf) (S i))) 
  (pos_Rl lf0 i)
Hreclf:ordered_Rlist lf ->
pos_Rl lf 0 = a ->
pos_Rl lf (Init.Nat.pred (Rlength lf)) = b ->
Rlength lf = S (Rlength lf0) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lf))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl lf i)
      (pos_Rl lf (S i))) 
   (pos_Rl lf0 i)) ->
b <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist lf lg)))
H8:In b (cons_ORlist (cons r lf) lg)
b <=
pos_Rl (cons_ORlist (cons r lf) lg)
  (Init.Nat.pred
     (Rlength (cons_ORlist (cons r lf) lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  apply RList_P7; [ apply RList_P2; assumption | assumption ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
Rlength (cons_ORlist lf lg) =
S (Rlength (FF (cons_ORlist lf lg) g))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  apply StepFun_P20; rewrite RList_P11; rewrite H7; rewrite H2; simpl;
    apply lt_O_Sn.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg0 i)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i) (pos_Rl lf (S i)))
  (pos_Rl lf0 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i)))
  (pos_Rl (FF (cons_ORlist lf lg) g) i)
  unfold constant_D_eq, open_interval; intros;
    cut
      (exists l : R,
        constant_D_eq g
        (open_interval (pos_Rl (cons_ORlist lf lg) i)
          (pos_Rl (cons_ORlist lf lg) (S i))) l).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
(exists l : R,
   constant_D_eq g
     (open_interval (pos_Rl (cons_ORlist lf lg) i)
        (pos_Rl (cons_ORlist lf lg) (S i))) l) ->
g x = pos_Rl (FF (cons_ORlist lf lg) g) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  intros; elim H11; clear H11; intros; assert (H12 := H11);
    assert
      (Hyp_cons :
        exists r : R, (exists r0 : Rlist, cons_ORlist lf lg = cons r r0)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
exists (r : R) (r0 : Rlist),
  cons_ORlist lf lg = cons r r0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
Hyp_cons:exists (r : R) (r0 : Rlist),
  cons_ORlist lf lg = cons r r0
g x = pos_Rl (FF (cons_ORlist lf lg) g) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply RList_P19; red; intro; rewrite H13 in H8; elim (lt_n_O _ H8).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
Hyp_cons:exists (r : R) (r0 : Rlist),
  cons_ORlist lf lg = cons r r0
g x = pos_Rl (FF (cons_ORlist lf lg) g) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  elim Hyp_cons; clear Hyp_cons; intros r [r0 Hyp_cons]; rewrite Hyp_cons;
    unfold FF; rewrite RList_P12.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
g x = g (pos_Rl (mid_Rlist r0 r) i)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  change (g x = g (pos_Rl (mid_Rlist (cons r r0) r) (S i)));
    rewrite <- Hyp_cons; rewrite RList_P13.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
g x =
g
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  assert (H13 := RList_P2 _ _ H _ H8); elim H13; intro.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
g x =
g
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i =
pos_Rl (cons_ORlist lf lg) (S i)
g x =
g
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  unfold constant_D_eq, open_interval in H11, H12; rewrite (H11 x H10);
    assert
      (H15 :
        pos_Rl (cons_ORlist lf lg) i <
        (pos_Rl (cons_ORlist lf lg) i + pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
        pos_Rl (cons_ORlist lf lg) (S i)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
g x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i <
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
g x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
H15:pos_Rl (cons_ORlist lf lg) i <
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
x0 =
g
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i =
pos_Rl (cons_ORlist lf lg) (S i)
g x =
g
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  split.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
g x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i <
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
g x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
g x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
H15:pos_Rl (cons_ORlist lf lg) i <
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
x0 =
g
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i =
pos_Rl (cons_ORlist lf lg) (S i)
g x =
g
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
            | discrR ] ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
g x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
g x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
H15:pos_Rl (cons_ORlist lf lg) i <
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
x0 =
g
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i =
pos_Rl (cons_ORlist lf lg) (S i)
g x =
g
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double;
            rewrite (Rplus_comm (pos_Rl (cons_ORlist lf lg) i));
              apply Rplus_lt_compat_l; assumption
            | discrR ] ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:forall x1 : R,
pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i) -> 
g x1 = x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
H15:pos_Rl (cons_ORlist lf lg) i <
(pos_Rl (cons_ORlist lf lg) i +
 pos_Rl (cons_ORlist lf lg) (S i)) / 2 <
pos_Rl (cons_ORlist lf lg) (S i)
x0 =
g
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i =
pos_Rl (cons_ORlist lf lg) (S i)
g x =
g
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  rewrite (H11 _ H15); reflexivity.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
H13:pos_Rl (cons_ORlist lf lg) i <=
pos_Rl (cons_ORlist lf lg) (S i)
H14:pos_Rl (cons_ORlist lf lg) i =
pos_Rl (cons_ORlist lf lg) (S i)
g x =
g
  ((pos_Rl (cons_ORlist lf lg) i +
    pos_Rl (cons_ORlist lf lg) (S i)) / 2)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  elim H10; intros; rewrite H14 in H15;
    elim (Rlt_irrefl _ (Rlt_trans _ _ _ H16 H15)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply H8.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
x0:R
H11, H12:constant_D_eq g
  (open_interval (pos_Rl (cons_ORlist lf lg) i)
     (pos_Rl (cons_ORlist lf lg) (S i))) x0
r:R
r0:Rlist
Hyp_cons:cons_ORlist lf lg = cons r r0
(i < Rlength (mid_Rlist r0 r))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  rewrite RList_P14; rewrite Hyp_cons in H8; simpl in H8; apply H8.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  assert (H11 : a < b).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
a < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
a <= pos_Rl (cons_ORlist lf lg) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  rewrite <- H6; rewrite <- (RList_P15 lf lg); try assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) 0 <=
pos_Rl (cons_ORlist lf lg) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf 0 = pos_Rl lg 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
ordered_Rlist (cons_ORlist lf lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(0 <= i)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(i < Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf 0 = pos_Rl lg 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply RList_P2; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(0 <= i)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(i < Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf 0 = pos_Rl lg 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply le_O_n.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(i < Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf 0 = pos_Rl lg 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply lt_trans with (pred (Rlength (cons_ORlist lf lg)));
    [ assumption
      | apply lt_pred_n_n; apply neq_O_lt; red; intro;
        rewrite <- H13 in H8; elim (lt_n_O _ H8) ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf 0 = pos_Rl lg 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  rewrite H1; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) (S i) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  elim H10; intros; apply Rlt_trans with x; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) (S i) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  rewrite <- H5; rewrite <- (RList_P16 lf lg); try assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl (cons_ORlist lf lg) (S i) <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf (Init.Nat.pred (Rlength lf)) =
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
ordered_Rlist (cons_ORlist lf lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(S i <= Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf (Init.Nat.pred (Rlength lf)) =
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply RList_P2; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(S i <= Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf (Init.Nat.pred (Rlength lf)) =
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply lt_n_Sm_le; apply lt_n_S; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H12:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf (Init.Nat.pred (Rlength lf)) =
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H13 in H8;
    elim (lt_n_O _ H8).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
pos_Rl lf (Init.Nat.pred (Rlength lf)) =
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  rewrite H0; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  set
    (I :=
      fun j:nat =>
        pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\ (j < Rlength lg)%nat);
    assert (H12 : Nbound I).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
Nbound I
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  unfold Nbound; exists (Rlength lg); intros; unfold I in H12; elim H12;
    intros; apply lt_le_weak; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  assert (H13 :  exists n : nat, I n).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
exists n : nat, I n
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  exists 0%nat; unfold I; split.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
pos_Rl lg 0 <= pos_Rl (cons_ORlist lf lg) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lg)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply Rle_trans with (pos_Rl (cons_ORlist lf lg) 0).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
pos_Rl lg 0 <= pos_Rl (cons_ORlist lf lg) 0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
pos_Rl (cons_ORlist lf lg) 0 <=
pos_Rl (cons_ORlist lf lg) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lg)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  right; symmetry ; rewrite H1; rewrite <- H6; apply RList_P15;
    try assumption; rewrite H1; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
pos_Rl (cons_ORlist lf lg) 0 <=
pos_Rl (cons_ORlist lf lg) i
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lg)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  elim (RList_P6 (cons_ORlist lf lg)); intros; apply H13;
    [ apply RList_P2; assumption
      | apply le_O_n
      | apply lt_trans with (pred (Rlength (cons_ORlist lf lg)));
        [ assumption
          | apply lt_pred_n_n; apply neq_O_lt; red; intro;
            rewrite <- H15 in H8; elim (lt_n_O _ H8) ] ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
(0 < Rlength lg)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  apply neq_O_lt; red; intro; rewrite <- H13 in H0;
    rewrite <- H1 in H11; rewrite <- H0 in H11; elim (Rlt_irrefl _ H11).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
exists l : R,
  constant_D_eq g
    (open_interval (pos_Rl (cons_ORlist lf lg) i)
       (pos_Rl (cons_ORlist lf lg) (S i))) l
  assert (H14 := Nzorn H13 H12); elim H14; clear H14; intros x0 H14;
    exists (pos_Rl lg0 x0); unfold constant_D_eq, open_interval;
      intros; assert (H16 := H4 x0); assert (H17 : (x0 < pred (Rlength lg))%nat).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
(x0 < Init.Nat.pred (Rlength lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  elim H14; clear H14; intros; unfold I in H14; elim H14; clear H14; intros;
    apply lt_S_n; replace (S (pred (Rlength lg))) with (Rlength lg).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
(S x0 < Rlength lg)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  inversion H18.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
(S x0 < S x0)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
m:nat
H20:(S x0 <= m)%nat
H19:S m = Rlength lg
(S x0 < S m)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  2: apply lt_n_S; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
(S x0 < S x0)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  cut (x0 = pred (Rlength lg)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
x0 = Init.Nat.pred (Rlength lg) -> (S x0 < S x0)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
x0 = Init.Nat.pred (Rlength lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  intro; rewrite H19 in H14; rewrite H0 in H14;
    cut (pos_Rl (cons_ORlist lf lg) i < b).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
pos_Rl (cons_ORlist lf lg) i < b -> (S x0 < S x0)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
x0 = Init.Nat.pred (Rlength lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H14 H21)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
pos_Rl (cons_ORlist lf lg) i < b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
x0 = Init.Nat.pred (Rlength lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
pos_Rl (cons_ORlist lf lg) i <
pos_Rl (cons_ORlist lf lg) (S i)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
pos_Rl (cons_ORlist lf lg) (S i) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
x0 = Init.Nat.pred (Rlength lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  elim H10; intros; apply Rlt_trans with x; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
pos_Rl (cons_ORlist lf lg) (S i) <= b
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
x0 = Init.Nat.pred (Rlength lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  rewrite <- H0;
    apply Rle_trans with
      (pos_Rl (cons_ORlist lf lg) (pred (Rlength (cons_ORlist lf lg)))).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
pos_Rl (cons_ORlist lf lg) (S i) <=
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg)))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) <=
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
x0 = Init.Nat.pred (Rlength lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  elim (RList_P6 (cons_ORlist lf lg)); intros; apply H21.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
H21:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H22:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
ordered_Rlist (cons_ORlist lf lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
H21:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H22:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(S i <= Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
H21:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H22:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) <=
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
x0 = Init.Nat.pred (Rlength lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  apply RList_P2; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
H21:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H22:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(S i <= Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
H21:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H22:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) <=
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
x0 = Init.Nat.pred (Rlength lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  apply lt_n_Sm_le; apply lt_n_S; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
H21:ordered_Rlist (cons_ORlist lf lg) ->
forall i0 j : nat,
(i0 <= j)%nat ->
(j < Rlength (cons_ORlist lf lg))%nat ->
pos_Rl (cons_ORlist lf lg) i0 <=
pos_Rl (cons_ORlist lf lg) j
H22:(forall i0 j : nat,
 (i0 <= j)%nat ->
 (j < Rlength (cons_ORlist lf lg))%nat ->
 pos_Rl (cons_ORlist lf lg) i0 <=
 pos_Rl (cons_ORlist lf lg) j) ->
ordered_Rlist (cons_ORlist lf lg)
(Init.Nat.pred (Rlength (cons_ORlist lf lg)) <
 Rlength (cons_ORlist lf lg))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) <=
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
x0 = Init.Nat.pred (Rlength lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H23 in H8;
    elim (lt_n_O _ H8).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
pos_Rl (cons_ORlist lf lg)
  (Init.Nat.pred (Rlength (cons_ORlist lf lg))) <=
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
x0 = Init.Nat.pred (Rlength lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  right; rewrite H0; rewrite <- H5; apply RList_P16; try assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:b <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
H19:x0 = Init.Nat.pred (Rlength lg)
pos_Rl lf (Init.Nat.pred (Rlength lf)) =
pos_Rl lg (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
x0 = Init.Nat.pred (Rlength lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  rewrite H0; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
H20:S x0 = Rlength lg
x0 = Init.Nat.pred (Rlength lg)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  rewrite <- H20; reflexivity.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H18:(x0 < Rlength lg)%nat
Rlength lg = S (Init.Nat.pred (Rlength lg))
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  apply S_pred with 0%nat; apply neq_O_lt; red; intro;
    rewrite <- H19 in H18; elim (lt_n_O _ H18).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
g x1 = pos_Rl lg0 x0
  assert (H18 := H16 H17); unfold constant_D_eq, open_interval in H18;
    rewrite (H18 x1).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
pos_Rl lg0 x0 = pos_Rl lg0 x0
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
pos_Rl lg x0 < x1 < pos_Rl lg (S x0)
  reflexivity.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
H14:I x0 /\
(forall i0 : nat, I i0 -> (i0 <= x0)%nat)
x1:R
H15:pos_Rl (cons_ORlist lf lg) i < x1 <
pos_Rl (cons_ORlist lf lg) (S i)
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
pos_Rl lg x0 < x1 < pos_Rl lg (S x0)
  elim H15; clear H15; intros; elim H14; clear H14; intros; unfold I in H14;
    elim H14; clear H14; intros; split.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
pos_Rl lg x0 < x1
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
x1 < pos_Rl lg (S x0)
  apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i); assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
x1 < pos_Rl lg (S x0)
  apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)); try assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lg (S x0)
  assert (H22 : (S x0 < Rlength lg)%nat).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
(S x0 < Rlength lg)%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
H22:(S x0 < Rlength lg)%nat
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lg (S x0)
  replace (Rlength lg) with (S (pred (Rlength lg))).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
(S x0 < S (Init.Nat.pred (Rlength lg)))%nat
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
S (Init.Nat.pred (Rlength lg)) = Rlength lg
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
H22:(S x0 < Rlength lg)%nat
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lg (S x0)
  apply lt_n_S; assumption.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
S (Init.Nat.pred (Rlength lg)) = Rlength lg
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
H22:(S x0 < Rlength lg)%nat
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lg (S x0)
  symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red;
    intro; rewrite <- H22 in H21; elim (lt_n_O _ H21).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
H22:(S x0 < Rlength lg)%nat
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lg (S x0)
  elim (Rle_dec (pos_Rl lg (S x0)) (pos_Rl (cons_ORlist lf lg) i)); intro a0.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
H22:(S x0 < Rlength lg)%nat
a0:pos_Rl lg (S x0) <= pos_Rl (cons_ORlist lf lg) i
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lg (S x0)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
H22:(S x0 < Rlength lg)%nat
a0:~
pos_Rl lg (S x0) <= pos_Rl (cons_ORlist lf lg) i
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lg (S x0)
  assert (H23 : (S x0 <= x0)%nat);
    [ apply H20; unfold I; split; assumption | elim (le_Sn_n _ H23) ].a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
H22:(S x0 < Rlength lg)%nat
a0:~
pos_Rl lg (S x0) <= pos_Rl (cons_ORlist lf lg) i
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lg (S x0)
  assert (H23 : pos_Rl (cons_ORlist lf lg) i < pos_Rl lg (S x0)).a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
H22:(S x0 < Rlength lg)%nat
a0:~
pos_Rl lg (S x0) <= pos_Rl (cons_ORlist lf lg) i
pos_Rl (cons_ORlist lf lg) i < pos_Rl lg (S x0)
a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
H22:(S x0 < Rlength lg)%nat
a0:~
pos_Rl lg (S x0) <= pos_Rl (cons_ORlist lf lg) i
H23:pos_Rl (cons_ORlist lf lg) i < pos_Rl lg (S x0)
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lg (S x0)
  auto with real.a, b:R
f, g:R -> R
lf, lg:Rlist
Hyp:a <= b
lg0, lf0:Rlist
Hyp_min:Rmin a b = a
Hyp_max:Rmax a b = b
H:ordered_Rlist lg
H1:pos_Rl lg 0 = a
H0:pos_Rl lg (Init.Nat.pred (Rlength lg)) = b
H2:Rlength lg = S (Rlength lg0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg0 i0)
H3:ordered_Rlist lf
H6:pos_Rl lf 0 = a
H5:pos_Rl lf (Init.Nat.pred (Rlength lf)) = b
H7:Rlength lf = S (Rlength lf0)
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lf))%nat ->
constant_D_eq f
  (open_interval (pos_Rl lf i0)
     (pos_Rl lf (S i0))) 
  (pos_Rl lf0 i0)
i:nat
H8:(i < Init.Nat.pred (Rlength (cons_ORlist lf lg)))%nat
x:R
H10:pos_Rl (cons_ORlist lf lg) i < x <
pos_Rl (cons_ORlist lf lg) (S i)
H11:a < b
I:=fun j : nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\
(j < Rlength lg)%nat:nat -> Prop
H12:Nbound I
H13:exists n : nat, I n
x0:nat
x1:R
H16:(x0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg x0)
     (pos_Rl lg (S x0))) 
  (pos_Rl lg0 x0)
H17:(x0 < Init.Nat.pred (Rlength lg))%nat
H18:forall x2 : R,
pos_Rl lg x0 < x2 < pos_Rl lg (S x0) ->
g x2 = pos_Rl lg0 x0
H15:pos_Rl (cons_ORlist lf lg) i < x1
H19:x1 < pos_Rl (cons_ORlist lf lg) (S i)
H20:forall i0 : nat, I i0 -> (i0 <= x0)%nat
H14:pos_Rl lg x0 <= pos_Rl (cons_ORlist lf lg) i
H21:(x0 < Rlength lg)%nat
H22:(S x0 < Rlength lg)%nat
a0:~
pos_Rl lg (S x0) <= pos_Rl (cons_ORlist lf lg) i
H23:pos_Rl (cons_ORlist lf lg) i < pos_Rl lg (S x0)
pos_Rl (cons_ORlist lf lg) (S i) <= pos_Rl lg (S x0)
  clear a0; apply RList_P17; try assumption;
    [ apply RList_P2; assumption
      | elim (RList_P9 lf lg (pos_Rl lg (S x0))); intros; apply H25; right;
        elim (RList_P3 lg (pos_Rl lg (S x0))); intros;
          apply H27; exists (S x0); split; [ reflexivity | apply H22 ] ].
Qed.

Lemma StepFun_P25 :
  forall (a b:R) (f g:R -> R) (lf lg:Rlist),
    is_subdivision f a b lf ->
    is_subdivision g a b lg -> is_subdivision g a b (cons_ORlist lf lg).forall (a b : R) (f g : R -> R) (lf lg : Rlist),
is_subdivision f a b lf ->
is_subdivision g a b lg ->
is_subdivision g a b (cons_ORlist lf lg)
Proof.forall (a b : R) (f g : R -> R) (lf lg : Rlist),
is_subdivision f a b lf ->
is_subdivision g a b lg ->
is_subdivision g a b (cons_ORlist lf lg)
  intros a b f g lf lg H H0; case (Rle_dec a b); intro;
    [ apply StepFun_P24 with f; assumption
      | apply StepFun_P5; apply StepFun_P24 with f;
        [ auto with real
          | apply StepFun_P5; assumption
          | apply StepFun_P5; assumption ] ].
Qed.

Lemma StepFun_P26 :
  forall (a b l:R) (f g:R -> R) (l1:Rlist),
    is_subdivision f a b l1 ->
    is_subdivision g a b l1 ->
    is_subdivision (fun x:R => f x + l * g x) a b l1.forall (a b l : R) (f g : R -> R) (l1 : Rlist),
is_subdivision f a b l1 ->
is_subdivision g a b l1 ->
is_subdivision (fun x : R => f x + l * g x) a b l1
Proof.forall (a b l : R) (f g : R -> R) (l1 : Rlist),
is_subdivision f a b l1 ->
is_subdivision g a b l1 ->
is_subdivision (fun x : R => f x + l * g x) a b l1
  intros a b l f g l1 (x0,(H0,(H1,(H2,(H3,H4)))))
    (x,(_,(_,(_,(_,H9))))).a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl x0 i)
x:Rlist
H9:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl x i)
is_subdivision (fun x1 : R => f x1 + l * g x1) a b l1
  exists (FF l1 (fun x:R => f x + l * g x)); repeat split; try assumption.a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl x0 i)
x:Rlist
H9:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl x i)
Rlength l1 =
S (Rlength (FF l1 (fun x1 : R => f x1 + l * g x1)))
a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl x0 i)
x:Rlist
H9:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl x i)
forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq (fun x1 : R => f x1 + l * g x1)
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl (FF l1 (fun x1 : R => f x1 + l * g x1)) i)
  apply StepFun_P20; rewrite H3; auto with arith.a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl x0 i)
x:Rlist
H9:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl x i)
forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq (fun x1 : R => f x1 + l * g x1)
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl (FF l1 (fun x1 : R => f x1 + l * g x1)) i)
  intros i H8 x1 H10; unfold open_interval in H10, H9, H4;
    rewrite (H9 _ H8 _ H10); rewrite (H4 _ H8 _ H10);
      assert (H11 : l1 <> nil).a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x0 i0)
x:Rlist
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x i0)
i:nat
H8:(i < Init.Nat.pred (Rlength l1))%nat
x1:R
H10:pos_Rl l1 i < x1 < pos_Rl l1 (S i)
l1 <> nil
a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x0 i0)
x:Rlist
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x i0)
i:nat
H8:(i < Init.Nat.pred (Rlength l1))%nat
x1:R
H10:pos_Rl l1 i < x1 < pos_Rl l1 (S i)
H11:l1 <> nil
pos_Rl x0 i + l * pos_Rl x i =
pos_Rl (FF l1 (fun x2 : R => f x2 + l * g x2)) i
  red; intro H11; rewrite H11 in H8; elim (lt_n_O _ H8).a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x0 i0)
x:Rlist
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x i0)
i:nat
H8:(i < Init.Nat.pred (Rlength l1))%nat
x1:R
H10:pos_Rl l1 i < x1 < pos_Rl l1 (S i)
H11:l1 <> nil
pos_Rl x0 i + l * pos_Rl x i =
pos_Rl (FF l1 (fun x2 : R => f x2 + l * g x2)) i
  destruct (RList_P19 _ H11) as (r,(r0,H12));
    rewrite H12; unfold FF;
      change
        (pos_Rl x0 i + l * pos_Rl x i =
          pos_Rl
          (app_Rlist (mid_Rlist (cons r r0) r) (fun x2:R => f x2 + l * g x2))
          (S i)); rewrite RList_P12.a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x0 i0)
x:Rlist
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x i0)
i:nat
H8:(i < Init.Nat.pred (Rlength l1))%nat
x1:R
H10:pos_Rl l1 i < x1 < pos_Rl l1 (S i)
H11:l1 <> nil
r:R
r0:Rlist
H12:l1 = cons r r0
pos_Rl x0 i + l * pos_Rl x i =
f (pos_Rl (mid_Rlist (cons r r0) r) (S i)) +
l * g (pos_Rl (mid_Rlist (cons r r0) r) (S i))
a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x0 i0)
x:Rlist
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x i0)
i:nat
H8:(i < Init.Nat.pred (Rlength l1))%nat
x1:R
H10:pos_Rl l1 i < x1 < pos_Rl l1 (S i)
H11:l1 <> nil
r:R
r0:Rlist
H12:l1 = cons r r0
(S i < Rlength (mid_Rlist (cons r r0) r))%nat
  rewrite RList_P13.a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x0 i0)
x:Rlist
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x i0)
i:nat
H8:(i < Init.Nat.pred (Rlength l1))%nat
x1:R
H10:pos_Rl l1 i < x1 < pos_Rl l1 (S i)
H11:l1 <> nil
r:R
r0:Rlist
H12:l1 = cons r r0
pos_Rl x0 i + l * pos_Rl x i =
f
  ((pos_Rl (cons r r0) i + pos_Rl (cons r r0) (S i)) /
   2) +
l *
g
  ((pos_Rl (cons r r0) i + pos_Rl (cons r r0) (S i)) /
   2)
a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x0 i0)
x:Rlist
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x i0)
i:nat
H8:(i < Init.Nat.pred (Rlength l1))%nat
x1:R
H10:pos_Rl l1 i < x1 < pos_Rl l1 (S i)
H11:l1 <> nil
r:R
r0:Rlist
H12:l1 = cons r r0
(i < Init.Nat.pred (Rlength (cons r r0)))%nat
a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x0 i0)
x:Rlist
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x i0)
i:nat
H8:(i < Init.Nat.pred (Rlength l1))%nat
x1:R
H10:pos_Rl l1 i < x1 < pos_Rl l1 (S i)
H11:l1 <> nil
r:R
r0:Rlist
H12:l1 = cons r r0
(S i < Rlength (mid_Rlist (cons r r0) r))%nat
  rewrite <- H12; rewrite (H9 _ H8); try rewrite (H4 _ H8);
    reflexivity ||
      (elim H10; clear H10; intros; split;
        [ apply Rmult_lt_reg_l with 2;
          [ prove_sup0
            | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
              rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
                [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l;
                  apply Rlt_trans with x1; assumption
                  | discrR ] ]
          | apply Rmult_lt_reg_l with 2;
            [ prove_sup0
              | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
                rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
                  [ rewrite Rmult_1_l; rewrite double;
                    rewrite (Rplus_comm (pos_Rl l1 i)); apply Rplus_lt_compat_l;
                      apply Rlt_trans with x1; assumption
                    | discrR ] ] ]).a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x0 i0)
x:Rlist
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x i0)
i:nat
H8:(i < Init.Nat.pred (Rlength l1))%nat
x1:R
H10:pos_Rl l1 i < x1 < pos_Rl l1 (S i)
H11:l1 <> nil
r:R
r0:Rlist
H12:l1 = cons r r0
(i < Init.Nat.pred (Rlength (cons r r0)))%nat
a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x0 i0)
x:Rlist
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x i0)
i:nat
H8:(i < Init.Nat.pred (Rlength l1))%nat
x1:R
H10:pos_Rl l1 i < x1 < pos_Rl l1 (S i)
H11:l1 <> nil
r:R
r0:Rlist
H12:l1 = cons r r0
(S i < Rlength (mid_Rlist (cons r r0) r))%nat
  rewrite <- H12; assumption.a, b, l:R
f, g:R -> R
l1, x0:Rlist
H0:ordered_Rlist l1
H1:pos_Rl l1 0 = Rmin a b
H2:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H3:Rlength l1 = S (Rlength x0)
H4:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x0 i0)
x:Rlist
H9:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq g
  (fun x2 : R =>
   pos_Rl l1 i0 < x2 < pos_Rl l1 (S i0))
  (pos_Rl x i0)
i:nat
H8:(i < Init.Nat.pred (Rlength l1))%nat
x1:R
H10:pos_Rl l1 i < x1 < pos_Rl l1 (S i)
H11:l1 <> nil
r:R
r0:Rlist
H12:l1 = cons r r0
(S i < Rlength (mid_Rlist (cons r r0) r))%nat
  rewrite RList_P14; simpl; rewrite H12 in H8; simpl in H8;
    apply lt_n_S; apply H8.
Qed.

Lemma StepFun_P27 :
  forall (a b l:R) (f g:R -> R) (lf lg:Rlist),
    is_subdivision f a b lf ->
    is_subdivision g a b lg ->
    is_subdivision (fun x:R => f x + l * g x) a b (cons_ORlist lf lg).forall (a b l : R) (f g : R -> R) (lf lg : Rlist),
is_subdivision f a b lf ->
is_subdivision g a b lg ->
is_subdivision (fun x : R => f x + l * g x) a b
  (cons_ORlist lf lg)
Proof.forall (a b l : R) (f g : R -> R) (lf lg : Rlist),
is_subdivision f a b lf ->
is_subdivision g a b lg ->
is_subdivision (fun x : R => f x + l * g x) a b
  (cons_ORlist lf lg)
  intros a b l f g lf lg H H0; apply StepFun_P26;
    [ apply StepFun_P23 with g; assumption
      | apply StepFun_P25 with f; assumption ].
Qed.

The set of step functions on a,b is a vectorial space

Lemma StepFun_P28 :
  forall (a b l:R) (f g:StepFun a b), IsStepFun (fun x:R => f x + l * g x) a b.forall (a b l : R) (f g : StepFun a b),
IsStepFun (fun x : R => f x + l * g x) a b
Proof.forall (a b l : R) (f g : StepFun a b),
IsStepFun (fun x : R => f x + l * g x) a b
  intros a b l f g; unfold IsStepFun; assert (H := pre f);
    assert (H0 := pre g); unfold IsStepFun in H, H0; elim H;
      elim H0; intros; apply existT with (cons_ORlist x0 x);
        apply StepFun_P27; assumption.
Qed.

Lemma StepFun_P29 :
  forall (a b:R) (f:StepFun a b), is_subdivision f a b (subdivision f).forall (a b : R) (f : StepFun a b),
is_subdivision f a b (subdivision f)
Proof.forall (a b : R) (f : StepFun a b),
is_subdivision f a b (subdivision f)
  intros a b f; unfold is_subdivision;
    apply existT with (subdivision_val f); apply StepFun_P1.
Qed.

Lemma StepFun_P30 :
  forall (a b l:R) (f g:StepFun a b),
    RiemannInt_SF (mkStepFun (StepFun_P28 l f g)) =
    RiemannInt_SF f + l * RiemannInt_SF g.forall (a b l : R) (f g : StepFun a b),
RiemannInt_SF
  {|
  fe := fun x : R => f x + l * g x;
  pre := StepFun_P28 l f g |} =
RiemannInt_SF f + l * RiemannInt_SF g
Proof.forall (a b l : R) (f g : StepFun a b),
RiemannInt_SF
  {|
  fe := fun x : R => f x + l * g x;
  pre := StepFun_P28 l f g |} =
RiemannInt_SF f + l * RiemannInt_SF g
  intros a b l f g; unfold RiemannInt_SF; case (Rle_dec a b);
    (intro;
      replace
      (Int_SF (subdivision_val (mkStepFun (StepFun_P28 l f g)))
        (subdivision (mkStepFun (StepFun_P28 l f g)))) with
      (Int_SF
        (FF (cons_ORlist (subdivision f) (subdivision g))
          (fun x:R => f x + l * g x))
        (cons_ORlist (subdivision f) (subdivision g)));
      [ rewrite StepFun_P19;
        replace
        (Int_SF (FF (cons_ORlist (subdivision f) (subdivision g)) f)
          (cons_ORlist (subdivision f) (subdivision g))) with
        (Int_SF (subdivision_val f) (subdivision f));
        [ replace
          (Int_SF (FF (cons_ORlist (subdivision f) (subdivision g)) g)
            (cons_ORlist (subdivision f) (subdivision g))) with
          (Int_SF (subdivision_val g) (subdivision g));
          [ ring
            | apply StepFun_P17 with (fe g) a b;
              [ apply StepFun_P1
                | apply StepFun_P21; apply StepFun_P25 with (fe f);
                  apply StepFun_P29 ] ]
          | apply StepFun_P17 with (fe f) a b;
            [ apply StepFun_P1
              | apply StepFun_P21; apply StepFun_P23 with (fe g);
                apply StepFun_P29 ] ]
        | apply StepFun_P17 with (fun x:R => f x + l * g x) a b;
          [ apply StepFun_P21; apply StepFun_P27; apply StepFun_P29
            | apply (StepFun_P1 (mkStepFun (StepFun_P28 l f g))) ] ]).
Qed.

Lemma StepFun_P31 :
  forall (a b:R) (f:R -> R) (l lf:Rlist),
    adapted_couple f a b l lf ->
    adapted_couple (fun x:R => Rabs (f x)) a b l (app_Rlist lf Rabs).forall (a b : R) (f : R -> R) (l lf : Rlist),
adapted_couple f a b l lf ->
adapted_couple (fun x : R => Rabs (f x)) a b l
  (app_Rlist lf Rabs)
Proof.forall (a b : R) (f : R -> R) (l lf : Rlist),
adapted_couple f a b l lf ->
adapted_couple (fun x : R => Rabs (f x)) a b l
  (app_Rlist lf Rabs)
  unfold adapted_couple; intros; decompose [and] H; clear H;
    repeat split; try assumption.a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
Rlength l = S (Rlength (app_Rlist lf Rabs))
a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq (fun x : R => Rabs (f x))
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl (app_Rlist lf Rabs) i)
  symmetry ; rewrite H3; rewrite RList_P18; reflexivity.a, b:R
f:R -> R
l, lf:Rlist
H0:ordered_Rlist l
H2:pos_Rl l 0 = Rmin a b
H1:pos_Rl l (Init.Nat.pred (Rlength l)) = Rmax a b
H3:Rlength l = S (Rlength lf)
H5:forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl lf i)
forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq (fun x : R => Rabs (f x))
  (open_interval (pos_Rl l i) (pos_Rl l (S i)))
  (pos_Rl (app_Rlist lf Rabs) i)
  intros; unfold constant_D_eq, open_interval;
    unfold constant_D_eq, open_interval in H5; intros;
      rewrite (H5 _ H _ H4); rewrite RList_P12;
        [ reflexivity | rewrite H3 in H; simpl in H; apply H ].
Qed.

Lemma StepFun_P32 :
  forall (a b:R) (f:StepFun a b), IsStepFun (fun x:R => Rabs (f x)) a b.forall (a b : R) (f : StepFun a b),
IsStepFun (fun x : R => Rabs (f x)) a b
Proof.forall (a b : R) (f : StepFun a b),
IsStepFun (fun x : R => Rabs (f x)) a b
  intros a b f; unfold IsStepFun; apply existT with (subdivision f);
    unfold is_subdivision;
      apply existT with (app_Rlist (subdivision_val f) Rabs);
        apply StepFun_P31; apply StepFun_P1.
Qed.

Lemma StepFun_P33 :
  forall l2 l1:Rlist,
    ordered_Rlist l1 -> Rabs (Int_SF l2 l1) <= Int_SF (app_Rlist l2 Rabs) l1.forall l2 l1 : Rlist,
ordered_Rlist l1 ->
Rabs (Int_SF l2 l1) <= Int_SF (app_Rlist l2 Rabs) l1
Proof.forall l2 l1 : Rlist,
ordered_Rlist l1 ->
Rabs (Int_SF l2 l1) <= Int_SF (app_Rlist l2 Rabs) l1
  simple induction l2; intros.l2, l1:Rlist
H:ordered_Rlist l1
Rabs (Int_SF nil l1) <= Int_SF (app_Rlist nil Rabs) l1
l2:Rlist
r:R
r0:Rlist
H:forall l0 : Rlist,
ordered_Rlist l0 ->
Rabs (Int_SF r0 l0) <=
Int_SF (app_Rlist r0 Rabs) l0
l1:Rlist
H0:ordered_Rlist l1
Rabs (Int_SF (cons r r0) l1) <=
Int_SF (app_Rlist (cons r r0) Rabs) l1
  simpl; rewrite Rabs_R0; right; reflexivity.l2:Rlist
r:R
r0:Rlist
H:forall l0 : Rlist,
ordered_Rlist l0 ->
Rabs (Int_SF r0 l0) <=
Int_SF (app_Rlist r0 Rabs) l0
l1:Rlist
H0:ordered_Rlist l1
Rabs (Int_SF (cons r r0) l1) <=
Int_SF (app_Rlist (cons r r0) Rabs) l1
  simpl; induction  l1 as [| r1 l1 Hrecl1].l2:Rlist
r:R
r0:Rlist
H:forall l1 : Rlist,
ordered_Rlist l1 ->
Rabs (Int_SF r0 l1) <=
Int_SF (app_Rlist r0 Rabs) l1
H0:ordered_Rlist nil
Rabs 0 <= 0
l2:Rlist
r:R
r0:Rlist
H:forall l0 : Rlist,
ordered_Rlist l0 ->
Rabs (Int_SF r0 l0) <=
Int_SF (app_Rlist r0 Rabs) l0
r1:R
l1:Rlist
H0:ordered_Rlist (cons r1 l1)
Hrecl1:ordered_Rlist l1 ->
Rabs
  match l1 with
  | nil => 0
  | cons x nil => 0
  | cons x (cons y k') =>
      r * (y - x) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons x nil => 0
| cons x (cons y k') =>
    Rabs r * (y - x) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
Rabs
  match l1 with
  | nil => 0
  | cons y k' => r * (y - r1) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons y k' =>
    Rabs r * (y - r1) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
  rewrite Rabs_R0; right; reflexivity.l2:Rlist
r:R
r0:Rlist
H:forall l0 : Rlist,
ordered_Rlist l0 ->
Rabs (Int_SF r0 l0) <=
Int_SF (app_Rlist r0 Rabs) l0
r1:R
l1:Rlist
H0:ordered_Rlist (cons r1 l1)
Hrecl1:ordered_Rlist l1 ->
Rabs
  match l1 with
  | nil => 0
  | cons x nil => 0
  | cons x (cons y k') =>
      r * (y - x) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons x nil => 0
| cons x (cons y k') =>
    Rabs r * (y - x) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
Rabs
  match l1 with
  | nil => 0
  | cons y k' => r * (y - r1) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons y k' =>
    Rabs r * (y - r1) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
  induction  l1 as [| r2 l1 Hrecl0].l2:Rlist
r:R
r0:Rlist
H:forall l1 : Rlist,
ordered_Rlist l1 ->
Rabs (Int_SF r0 l1) <=
Int_SF (app_Rlist r0 Rabs) l1
r1:R
H0:ordered_Rlist (cons r1 nil)
Hrecl1:ordered_Rlist nil -> Rabs 0 <= 0
Rabs 0 <= 0
l2:Rlist
r:R
r0:Rlist
H:forall l0 : Rlist,
ordered_Rlist l0 ->
Rabs (Int_SF r0 l0) <=
Int_SF (app_Rlist r0 Rabs) l0
r1, r2:R
l1:Rlist
H0:ordered_Rlist (cons r1 (cons r2 l1))
Hrecl1:ordered_Rlist (cons r2 l1) ->
Rabs
  match l1 with
  | nil => 0
  | cons y k' =>
      r * (y - r2) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons y k' =>
    Rabs r * (y - r2) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
Hrecl0:ordered_Rlist (cons r1 l1) ->
(ordered_Rlist l1 ->
 Rabs
   match l1 with
   | nil => 0
   | cons x nil => 0
   | cons x (cons y k') =>
       r * (y - x) + Int_SF r0 (cons y k')
   end <=
 match l1 with
 | nil => 0
 | cons x nil => 0
 | cons x (cons y k') =>
     Rabs r * (y - x) +
     Int_SF (app_Rlist r0 Rabs) (cons y k')
 end) ->
Rabs
  match l1 with
  | nil => 0
  | cons y k' =>
      r * (y - r1) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons y k' =>
    Rabs r * (y - r1) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
Rabs (r * (r2 - r1) + Int_SF r0 (cons r2 l1)) <=
Rabs r * (r2 - r1) +
Int_SF (app_Rlist r0 Rabs) (cons r2 l1)
  rewrite Rabs_R0; right; reflexivity.l2:Rlist
r:R
r0:Rlist
H:forall l0 : Rlist,
ordered_Rlist l0 ->
Rabs (Int_SF r0 l0) <=
Int_SF (app_Rlist r0 Rabs) l0
r1, r2:R
l1:Rlist
H0:ordered_Rlist (cons r1 (cons r2 l1))
Hrecl1:ordered_Rlist (cons r2 l1) ->
Rabs
  match l1 with
  | nil => 0
  | cons y k' =>
      r * (y - r2) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons y k' =>
    Rabs r * (y - r2) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
Hrecl0:ordered_Rlist (cons r1 l1) ->
(ordered_Rlist l1 ->
 Rabs
   match l1 with
   | nil => 0
   | cons x nil => 0
   | cons x (cons y k') =>
       r * (y - x) + Int_SF r0 (cons y k')
   end <=
 match l1 with
 | nil => 0
 | cons x nil => 0
 | cons x (cons y k') =>
     Rabs r * (y - x) +
     Int_SF (app_Rlist r0 Rabs) (cons y k')
 end) ->
Rabs
  match l1 with
  | nil => 0
  | cons y k' =>
      r * (y - r1) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons y k' =>
    Rabs r * (y - r1) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
Rabs (r * (r2 - r1) + Int_SF r0 (cons r2 l1)) <=
Rabs r * (r2 - r1) +
Int_SF (app_Rlist r0 Rabs) (cons r2 l1)
  apply Rle_trans with (Rabs (r * (r2 - r1)) + Rabs (Int_SF r0 (cons r2 l1))).l2:Rlist
r:R
r0:Rlist
H:forall l0 : Rlist,
ordered_Rlist l0 ->
Rabs (Int_SF r0 l0) <=
Int_SF (app_Rlist r0 Rabs) l0
r1, r2:R
l1:Rlist
H0:ordered_Rlist (cons r1 (cons r2 l1))
Hrecl1:ordered_Rlist (cons r2 l1) ->
Rabs
  match l1 with
  | nil => 0
  | cons y k' =>
      r * (y - r2) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons y k' =>
    Rabs r * (y - r2) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
Hrecl0:ordered_Rlist (cons r1 l1) ->
(ordered_Rlist l1 ->
 Rabs
   match l1 with
   | nil => 0
   | cons x nil => 0
   | cons x (cons y k') =>
       r * (y - x) + Int_SF r0 (cons y k')
   end <=
 match l1 with
 | nil => 0
 | cons x nil => 0
 | cons x (cons y k') =>
     Rabs r * (y - x) +
     Int_SF (app_Rlist r0 Rabs) (cons y k')
 end) ->
Rabs
  match l1 with
  | nil => 0
  | cons y k' =>
      r * (y - r1) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons y k' =>
    Rabs r * (y - r1) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
Rabs (r * (r2 - r1) + Int_SF r0 (cons r2 l1)) <=
Rabs (r * (r2 - r1)) + Rabs (Int_SF r0 (cons r2 l1))
l2:Rlist
r:R
r0:Rlist
H:forall l0 : Rlist,
ordered_Rlist l0 ->
Rabs (Int_SF r0 l0) <=
Int_SF (app_Rlist r0 Rabs) l0
r1, r2:R
l1:Rlist
H0:ordered_Rlist (cons r1 (cons r2 l1))
Hrecl1:ordered_Rlist (cons r2 l1) ->
Rabs
  match l1 with
  | nil => 0
  | cons y k' =>
      r * (y - r2) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons y k' =>
    Rabs r * (y - r2) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
Hrecl0:ordered_Rlist (cons r1 l1) ->
(ordered_Rlist l1 ->
 Rabs
   match l1 with
   | nil => 0
   | cons x nil => 0
   | cons x (cons y k') =>
       r * (y - x) + Int_SF r0 (cons y k')
   end <=
 match l1 with
 | nil => 0
 | cons x nil => 0
 | cons x (cons y k') =>
     Rabs r * (y - x) +
     Int_SF (app_Rlist r0 Rabs) (cons y k')
 end) ->
Rabs
  match l1 with
  | nil => 0
  | cons y k' =>
      r * (y - r1) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons y k' =>
    Rabs r * (y - r1) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
Rabs (r * (r2 - r1)) + Rabs (Int_SF r0 (cons r2 l1)) <=
Rabs r * (r2 - r1) +
Int_SF (app_Rlist r0 Rabs) (cons r2 l1)
  apply Rabs_triang.l2:Rlist
r:R
r0:Rlist
H:forall l0 : Rlist,
ordered_Rlist l0 ->
Rabs (Int_SF r0 l0) <=
Int_SF (app_Rlist r0 Rabs) l0
r1, r2:R
l1:Rlist
H0:ordered_Rlist (cons r1 (cons r2 l1))
Hrecl1:ordered_Rlist (cons r2 l1) ->
Rabs
  match l1 with
  | nil => 0
  | cons y k' =>
      r * (y - r2) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons y k' =>
    Rabs r * (y - r2) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
Hrecl0:ordered_Rlist (cons r1 l1) ->
(ordered_Rlist l1 ->
 Rabs
   match l1 with
   | nil => 0
   | cons x nil => 0
   | cons x (cons y k') =>
       r * (y - x) + Int_SF r0 (cons y k')
   end <=
 match l1 with
 | nil => 0
 | cons x nil => 0
 | cons x (cons y k') =>
     Rabs r * (y - x) +
     Int_SF (app_Rlist r0 Rabs) (cons y k')
 end) ->
Rabs
  match l1 with
  | nil => 0
  | cons y k' =>
      r * (y - r1) + Int_SF r0 (cons y k')
  end <=
match l1 with
| nil => 0
| cons y k' =>
    Rabs r * (y - r1) +
    Int_SF (app_Rlist r0 Rabs) (cons y k')
end
Rabs (r * (r2 - r1)) + Rabs (Int_SF r0 (cons r2 l1)) <=
Rabs r * (r2 - r1) +
Int_SF (app_Rlist r0 Rabs) (cons r2 l1)
  rewrite Rabs_mult; rewrite (Rabs_right (r2 - r1));
    [ apply Rplus_le_compat_l; apply H; apply RList_P4 with r1; assumption
      | apply Rge_minus; apply Rle_ge; apply (H0 0%nat); simpl;
        apply lt_O_Sn ].
Qed.

Lemma StepFun_P34 :
  forall (a b:R) (f:StepFun a b),
    a <= b ->
    Rabs (RiemannInt_SF f) <= RiemannInt_SF (mkStepFun (StepFun_P32 f)).forall (a b : R) (f : StepFun a b),
a <= b ->
Rabs (RiemannInt_SF f) <=
RiemannInt_SF
  {|
  fe := fun x : R => Rabs (f x);
  pre := StepFun_P32 f |}
Proof.forall (a b : R) (f : StepFun a b),
a <= b ->
Rabs (RiemannInt_SF f) <=
RiemannInt_SF
  {|
  fe := fun x : R => Rabs (f x);
  pre := StepFun_P32 f |}
  intros; unfold RiemannInt_SF; decide (Rle_dec a b) with H.a, b:R
f:StepFun a b
H:a <= b
Rabs (Int_SF (subdivision_val f) (subdivision f)) <=
Int_SF
  (subdivision_val
     {|
     fe := fun x : R => Rabs (f x);
     pre := StepFun_P32 f |})
  (subdivision
     {|
     fe := fun x : R => Rabs (f x);
     pre := StepFun_P32 f |})
  replace
  (Int_SF (subdivision_val (mkStepFun (StepFun_P32 f)))
    (subdivision (mkStepFun (StepFun_P32 f)))) with
  (Int_SF (app_Rlist (subdivision_val f) Rabs) (subdivision f)).a, b:R
f:StepFun a b
H:a <= b
Rabs (Int_SF (subdivision_val f) (subdivision f)) <=
Int_SF (app_Rlist (subdivision_val f) Rabs)
  (subdivision f)
a, b:R
f:StepFun a b
H:a <= b
Int_SF (app_Rlist (subdivision_val f) Rabs)
  (subdivision f) =
Int_SF
  (subdivision_val
     {|
     fe := fun x : R => Rabs (f x);
     pre := StepFun_P32 f |})
  (subdivision
     {|
     fe := fun x : R => Rabs (f x);
     pre := StepFun_P32 f |})
  apply StepFun_P33; assert (H0 := StepFun_P29 f); unfold is_subdivision in H0;
    elim H0; intros; unfold adapted_couple in p; decompose [and] p;
      assumption.a, b:R
f:StepFun a b
H:a <= b
Int_SF (app_Rlist (subdivision_val f) Rabs)
  (subdivision f) =
Int_SF
  (subdivision_val
     {|
     fe := fun x : R => Rabs (f x);
     pre := StepFun_P32 f |})
  (subdivision
     {|
     fe := fun x : R => Rabs (f x);
     pre := StepFun_P32 f |})
  apply StepFun_P17 with (fun x:R => Rabs (f x)) a b;
    [ apply StepFun_P31; apply StepFun_P1
      | apply (StepFun_P1 (mkStepFun (StepFun_P32 f))) ].
Qed.

Lemma StepFun_P35 :
  forall (l:Rlist) (a b:R) (f g:R -> R),
    ordered_Rlist l ->
    pos_Rl l 0 = a ->
    pos_Rl l (pred (Rlength l)) = b ->
    (forall x:R, a < x < b -> f x <= g x) ->
    Int_SF (FF l f) l <= Int_SF (FF l g) l.forall (l : Rlist) (a b : R) (f g : R -> R),
ordered_Rlist l ->
pos_Rl l 0 = a ->
pos_Rl l (Init.Nat.pred (Rlength l)) = b ->
(forall x : R, a < x < b -> f x <= g x) ->
Int_SF (FF l f) l <= Int_SF (FF l g) l
Proof.forall (l : Rlist) (a b : R) (f g : R -> R),
ordered_Rlist l ->
pos_Rl l 0 = a ->
pos_Rl l (Init.Nat.pred (Rlength l)) = b ->
(forall x : R, a < x < b -> f x <= g x) ->
Int_SF (FF l f) l <= Int_SF (FF l g) l
  simple induction l; intros.l:Rlist
a, b:R
f, g:R -> R
H:ordered_Rlist nil
H0:pos_Rl nil 0 = a
H1:pos_Rl nil (Init.Nat.pred (Rlength nil)) = b
H2:forall x : R, a < x < b -> f x <= g x
Int_SF (FF nil f) nil <= Int_SF (FF nil g) nil
l:Rlist
r:R
r0:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist r0 ->
pos_Rl r0 0 = a0 ->
pos_Rl r0 (Init.Nat.pred (Rlength r0)) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF r0 f0) r0 <= Int_SF (FF r0 g0) r0
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r r0)
H1:pos_Rl (cons r r0) 0 = a
H2:pos_Rl (cons r r0)
  (Init.Nat.pred (Rlength (cons r r0))) = b
H3:forall x : R, a < x < b -> f x <= g x
Int_SF (FF (cons r r0) f) (cons r r0) <=
Int_SF (FF (cons r r0) g) (cons r r0)
  right; reflexivity.l:Rlist
r:R
r0:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist r0 ->
pos_Rl r0 0 = a0 ->
pos_Rl r0 (Init.Nat.pred (Rlength r0)) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF r0 f0) r0 <= Int_SF (FF r0 g0) r0
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r r0)
H1:pos_Rl (cons r r0) 0 = a
H2:pos_Rl (cons r r0)
  (Init.Nat.pred (Rlength (cons r r0))) = b
H3:forall x : R, a < x < b -> f x <= g x
Int_SF (FF (cons r r0) f) (cons r r0) <=
Int_SF (FF (cons r r0) g) (cons r r0)
  simpl; induction  r0 as [| r0 r1 Hrecr0].l:Rlist
r:R
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist nil ->
pos_Rl nil 0 = a0 ->
pos_Rl nil (Init.Nat.pred (Rlength nil)) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF nil f0) nil <= Int_SF (FF nil g0) nil
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r nil)
H1:pos_Rl (cons r nil) 0 = a
H2:pos_Rl (cons r nil)
  (Init.Nat.pred (Rlength (cons r nil))) = b
H3:forall x : R, a < x < b -> f x <= g x
Int_SF (app_Rlist (mid_Rlist nil r) f) (cons r nil) <=
Int_SF (app_Rlist (mid_Rlist nil r) g) (cons r nil)
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist (cons r0 r1) r) f)
  (cons r (cons r0 r1)) <=
Int_SF (app_Rlist (mid_Rlist (cons r0 r1) r) g)
  (cons r (cons r0 r1))
  right; reflexivity.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist (cons r0 r1) r) f)
  (cons r (cons r0 r1)) <=
Int_SF (app_Rlist (mid_Rlist (cons r0 r1) r) g)
  (cons r (cons r0 r1))
  simpl; apply Rplus_le_compat.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
f ((r + r0) / 2) * (r0 - r) <=
g ((r + r0) / 2) * (r0 - r)
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  case (Req_dec r r0); intro.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r = r0
f ((r + r0) / 2) * (r0 - r) <=
g ((r + r0) / 2) * (r0 - r)
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
f ((r + r0) / 2) * (r0 - r) <=
g ((r + r0) / 2) * (r0 - r)
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  rewrite H4; right; ring.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
f ((r + r0) / 2) * (r0 - r) <=
g ((r + r0) / 2) * (r0 - r)
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  do 2 rewrite <- (Rmult_comm (r0 - r)); apply Rmult_le_compat_l.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
0 <= r0 - r
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
f ((r + r0) / 2) <= g ((r + r0) / 2)
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  apply Rge_le; apply Rge_minus; apply Rle_ge; apply (H0 0%nat); simpl;
    apply lt_O_Sn.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
f ((r + r0) / 2) <= g ((r + r0) / 2)
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  apply H3; split.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
a < (r + r0) / 2
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
(r + r0) / 2 < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  apply Rmult_lt_reg_l with 2.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
0 < 2
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 * a < 2 * ((r + r0) / 2)
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
(r + r0) / 2 < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  prove_sup0.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 * a < 2 * ((r + r0) / 2)
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
(r + r0) / 2 < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
    rewrite <- Rinv_r_sym.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 * a < 1 * (r + r0)
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
(r + r0) / 2 < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  assert (H5 : r = a).l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
r = a
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r = a
2 * a < 1 * (r + r0)
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
(r + r0) / 2 < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  apply H1.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r = a
2 * a < 1 * (r + r0)
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
(r + r0) / 2 < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  rewrite H5; rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r = a
a < r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
(r + r0) / 2 < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  assert (H6 := H0 0%nat (lt_O_Sn _)).l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r = a
H6:pos_Rl (cons r (cons r0 r1)) 0 <=
pos_Rl (cons r (cons r0 r1)) 1
a < r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
(r + r0) / 2 < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  simpl in H6.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r = a
H6:r <= r0
a < r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
(r + r0) / 2 < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  elim H6; intro.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r = a
H6:r <= r0
H7:r < r0
a < r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r = a
H6:r <= r0
H7:r = r0
a < r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
(r + r0) / 2 < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  rewrite H5 in H7; apply H7.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r = a
H6:r <= r0
H7:r = r0
a < r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
(r + r0) / 2 < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  elim H4; assumption.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
(r + r0) / 2 < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  discrR.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
(r + r0) / 2 < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  apply Rmult_lt_reg_l with 2.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
0 < 2
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 * ((r + r0) / 2) < 2 * b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  prove_sup0.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 * ((r + r0) / 2) < 2 * b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
    rewrite <- Rinv_r_sym.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
1 * (r + r0) < 2 * b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  rewrite Rmult_1_l; rewrite double; assert (H5 : r0 <= b).l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
r0 <= b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r + r0 < b + b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  replace b with
  (pos_Rl (cons r (cons r0 r1)) (pred (Rlength (cons r (cons r0 r1))))).l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
r0 <=
pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1))))
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r + r0 < b + b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  replace r0 with (pos_Rl (cons r (cons r0 r1)) 1).l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
pos_Rl (cons r (cons r0 r1)) 1 <=
pos_Rl
  (cons r (cons (pos_Rl (cons r (cons r0 r1)) 1) r1))
  (Init.Nat.pred
     (Rlength
        (cons r
           (cons (pos_Rl (cons r (cons r0 r1)) 1) r1))))
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
pos_Rl (cons r (cons r0 r1)) 1 = r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r + r0 < b + b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  elim (RList_P6 (cons r (cons r0 r1))); intros; apply H5.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:ordered_Rlist (cons r (cons r0 r1)) ->
forall i j : nat,
(i <= j)%nat ->
(j < Rlength (cons r (cons r0 r1)))%nat ->
pos_Rl (cons r (cons r0 r1)) i <=
pos_Rl (cons r (cons r0 r1)) j
H6:(forall i j : nat,
 (i <= j)%nat ->
 (j < Rlength (cons r (cons r0 r1)))%nat ->
 pos_Rl (cons r (cons r0 r1)) i <=
 pos_Rl (cons r (cons r0 r1)) j) ->
ordered_Rlist (cons r (cons r0 r1))
ordered_Rlist (cons r (cons r0 r1))
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:ordered_Rlist (cons r (cons r0 r1)) ->
forall i j : nat,
(i <= j)%nat ->
(j < Rlength (cons r (cons r0 r1)))%nat ->
pos_Rl (cons r (cons r0 r1)) i <=
pos_Rl (cons r (cons r0 r1)) j
H6:(forall i j : nat,
 (i <= j)%nat ->
 (j < Rlength (cons r (cons r0 r1)))%nat ->
 pos_Rl (cons r (cons r0 r1)) i <=
 pos_Rl (cons r (cons r0 r1)) j) ->
ordered_Rlist (cons r (cons r0 r1))
(1 <=
 Init.Nat.pred
   (Rlength
      (cons r
         (cons (pos_Rl (cons r (cons r0 r1)) 1) r1))))%nat
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:ordered_Rlist (cons r (cons r0 r1)) ->
forall i j : nat,
(i <= j)%nat ->
(j < Rlength (cons r (cons r0 r1)))%nat ->
pos_Rl (cons r (cons r0 r1)) i <=
pos_Rl (cons r (cons r0 r1)) j
H6:(forall i j : nat,
 (i <= j)%nat ->
 (j < Rlength (cons r (cons r0 r1)))%nat ->
 pos_Rl (cons r (cons r0 r1)) i <=
 pos_Rl (cons r (cons r0 r1)) j) ->
ordered_Rlist (cons r (cons r0 r1))
(Init.Nat.pred
   (Rlength
      (cons r
         (cons (pos_Rl (cons r (cons r0 r1)) 1) r1))) <
 Rlength (cons r (cons r0 r1)))%nat
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
pos_Rl (cons r (cons r0 r1)) 1 = r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r + r0 < b + b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  assumption.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:ordered_Rlist (cons r (cons r0 r1)) ->
forall i j : nat,
(i <= j)%nat ->
(j < Rlength (cons r (cons r0 r1)))%nat ->
pos_Rl (cons r (cons r0 r1)) i <=
pos_Rl (cons r (cons r0 r1)) j
H6:(forall i j : nat,
 (i <= j)%nat ->
 (j < Rlength (cons r (cons r0 r1)))%nat ->
 pos_Rl (cons r (cons r0 r1)) i <=
 pos_Rl (cons r (cons r0 r1)) j) ->
ordered_Rlist (cons r (cons r0 r1))
(1 <=
 Init.Nat.pred
   (Rlength
      (cons r
         (cons (pos_Rl (cons r (cons r0 r1)) 1) r1))))%nat
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:ordered_Rlist (cons r (cons r0 r1)) ->
forall i j : nat,
(i <= j)%nat ->
(j < Rlength (cons r (cons r0 r1)))%nat ->
pos_Rl (cons r (cons r0 r1)) i <=
pos_Rl (cons r (cons r0 r1)) j
H6:(forall i j : nat,
 (i <= j)%nat ->
 (j < Rlength (cons r (cons r0 r1)))%nat ->
 pos_Rl (cons r (cons r0 r1)) i <=
 pos_Rl (cons r (cons r0 r1)) j) ->
ordered_Rlist (cons r (cons r0 r1))
(Init.Nat.pred
   (Rlength
      (cons r
         (cons (pos_Rl (cons r (cons r0 r1)) 1) r1))) <
 Rlength (cons r (cons r0 r1)))%nat
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
pos_Rl (cons r (cons r0 r1)) 1 = r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r + r0 < b + b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  simpl; apply le_n_S.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:ordered_Rlist (cons r (cons r0 r1)) ->
forall i j : nat,
(i <= j)%nat ->
(j < Rlength (cons r (cons r0 r1)))%nat ->
pos_Rl (cons r (cons r0 r1)) i <=
pos_Rl (cons r (cons r0 r1)) j
H6:(forall i j : nat,
 (i <= j)%nat ->
 (j < Rlength (cons r (cons r0 r1)))%nat ->
 pos_Rl (cons r (cons r0 r1)) i <=
 pos_Rl (cons r (cons r0 r1)) j) ->
ordered_Rlist (cons r (cons r0 r1))
(0 <= Rlength r1)%nat
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:ordered_Rlist (cons r (cons r0 r1)) ->
forall i j : nat,
(i <= j)%nat ->
(j < Rlength (cons r (cons r0 r1)))%nat ->
pos_Rl (cons r (cons r0 r1)) i <=
pos_Rl (cons r (cons r0 r1)) j
H6:(forall i j : nat,
 (i <= j)%nat ->
 (j < Rlength (cons r (cons r0 r1)))%nat ->
 pos_Rl (cons r (cons r0 r1)) i <=
 pos_Rl (cons r (cons r0 r1)) j) ->
ordered_Rlist (cons r (cons r0 r1))
(Init.Nat.pred
   (Rlength
      (cons r
         (cons (pos_Rl (cons r (cons r0 r1)) 1) r1))) <
 Rlength (cons r (cons r0 r1)))%nat
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
pos_Rl (cons r (cons r0 r1)) 1 = r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r + r0 < b + b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  apply le_O_n.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:ordered_Rlist (cons r (cons r0 r1)) ->
forall i j : nat,
(i <= j)%nat ->
(j < Rlength (cons r (cons r0 r1)))%nat ->
pos_Rl (cons r (cons r0 r1)) i <=
pos_Rl (cons r (cons r0 r1)) j
H6:(forall i j : nat,
 (i <= j)%nat ->
 (j < Rlength (cons r (cons r0 r1)))%nat ->
 pos_Rl (cons r (cons r0 r1)) i <=
 pos_Rl (cons r (cons r0 r1)) j) ->
ordered_Rlist (cons r (cons r0 r1))
(Init.Nat.pred
   (Rlength
      (cons r
         (cons (pos_Rl (cons r (cons r0 r1)) 1) r1))) <
 Rlength (cons r (cons r0 r1)))%nat
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
pos_Rl (cons r (cons r0 r1)) 1 = r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r + r0 < b + b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  simpl; apply lt_n_Sn.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
pos_Rl (cons r (cons r0 r1)) 1 = r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r + r0 < b + b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  reflexivity.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r + r0 < b + b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  apply Rle_lt_trans with (r + b).l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r + r0 <= r + b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r + b < b + b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  apply Rplus_le_compat_l; assumption.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r + b < b + b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  rewrite (Rplus_comm r); apply Rplus_lt_compat_l.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r < b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  apply Rlt_le_trans with r0.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r < r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r0 <= b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  assert (H6 := H0 0%nat (lt_O_Sn _)).l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
H6:pos_Rl (cons r (cons r0 r1)) 0 <=
pos_Rl (cons r (cons r0 r1)) 1
r < r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r0 <= b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  simpl in H6.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
H6:r <= r0
r < r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r0 <= b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  elim H6; intro.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
H6:r <= r0
H7:r < r0
r < r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
H6:r <= r0
H7:r = r0
r < r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r0 <= b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  apply H7.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
H6:r <= r0
H7:r = r0
r < r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r0 <= b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  elim H4; assumption.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
H5:r0 <= b
r0 <= b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  assumption.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
H4:r <> r0
2 <> 0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  discrR.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
pos_Rl (cons r0 r1) 0 = a0 ->
pos_Rl (cons r0 r1)
  (Init.Nat.pred (Rlength (cons r0 r1))) = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (FF (cons r0 r1) f0) (cons r0 r1) <=
Int_SF (FF (cons r0 r1) g0) (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
Int_SF (app_Rlist (mid_Rlist r1 r0) f) (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g) (cons r0 r1)
  simpl in H; apply H with r0 b.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
r0 = a0 ->
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (app_Rlist (mid_Rlist r1 r0) f0)
  (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g0)
  (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
ordered_Rlist (cons r0 r1)
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
r0 = a0 ->
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (app_Rlist (mid_Rlist r1 r0) f0)
  (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g0)
  (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
r0 = r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
r0 = a0 ->
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (app_Rlist (mid_Rlist r1 r0) f0)
  (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g0)
  (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
r0 = a0 ->
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (app_Rlist (mid_Rlist r1 r0) f0)
  (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g0)
  (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
forall x : R, r0 < x < b -> f x <= g x
  apply RList_P4 with r; assumption.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
r0 = a0 ->
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (app_Rlist (mid_Rlist r1 r0) f0)
  (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g0)
  (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
r0 = r0
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
r0 = a0 ->
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (app_Rlist (mid_Rlist r1 r0) f0)
  (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g0)
  (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
r0 = a0 ->
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (app_Rlist (mid_Rlist r1 r0) f0)
  (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g0)
  (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
forall x : R, r0 < x < b -> f x <= g x
  reflexivity.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
r0 = a0 ->
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (app_Rlist (mid_Rlist r1 r0) f0)
  (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g0)
  (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b
l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
r0 = a0 ->
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (app_Rlist (mid_Rlist r1 r0) f0)
  (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g0)
  (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
forall x : R, r0 < x < b -> f x <= g x
  rewrite <- H2; reflexivity.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
r0 = a0 ->
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b0 ->
(forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
Int_SF (app_Rlist (mid_Rlist r1 r0) f0)
  (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g0)
  (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x : R, a < x < b -> f x <= g x
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x : R, a0 < x < b0 -> f0 x <= g0 x) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
forall x : R, r0 < x < b -> f x <= g x
  intros; apply H3; elim H4; intros; split; try assumption.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
r0 = a0 ->
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b0 ->
(forall x0 : R, a0 < x0 < b0 -> f0 x0 <= g0 x0) ->
Int_SF (app_Rlist (mid_Rlist r1 r0) f0)
  (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g0)
  (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x0 : R, a < x0 < b -> f x0 <= g x0
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x0 : R,
  a0 < x0 < b0 -> f0 x0 <= g0 x0) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
x:R
H4:r0 < x < b
H5:r0 < x
H6:x < b
a < x
  apply Rle_lt_trans with r0; try assumption.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
r0 = a0 ->
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b0 ->
(forall x0 : R, a0 < x0 < b0 -> f0 x0 <= g0 x0) ->
Int_SF (app_Rlist (mid_Rlist r1 r0) f0)
  (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g0)
  (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x0 : R, a < x0 < b -> f x0 <= g x0
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x0 : R,
  a0 < x0 < b0 -> f0 x0 <= g0 x0) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
x:R
H4:r0 < x < b
H5:r0 < x
H6:x < b
a <= r0
  rewrite <- H1.l:Rlist
r, r0:R
r1:Rlist
H:forall (a0 b0 : R) (f0 g0 : R -> R),
ordered_Rlist (cons r0 r1) ->
r0 = a0 ->
match Rlength r1 with
| 0%nat => r0
| S i' => pos_Rl r1 i'
end = b0 ->
(forall x0 : R, a0 < x0 < b0 -> f0 x0 <= g0 x0) ->
Int_SF (app_Rlist (mid_Rlist r1 r0) f0)
  (cons r0 r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r0) g0)
  (cons r0 r1)
a, b:R
f, g:R -> R
H0:ordered_Rlist (cons r (cons r0 r1))
H1:pos_Rl (cons r (cons r0 r1)) 0 = a
H2:pos_Rl (cons r (cons r0 r1))
  (Init.Nat.pred (Rlength (cons r (cons r0 r1)))) =
b
H3:forall x0 : R, a < x0 < b -> f x0 <= g x0
Hrecr0:(forall (a0 b0 : R) (f0 g0 : R -> R),
 ordered_Rlist r1 ->
 pos_Rl r1 0 = a0 ->
 pos_Rl r1 (Init.Nat.pred (Rlength r1)) = b0 ->
 (forall x0 : R,
  a0 < x0 < b0 -> f0 x0 <= g0 x0) ->
 Int_SF (FF r1 f0) r1 <= Int_SF (FF r1 g0) r1) ->
ordered_Rlist (cons r r1) ->
pos_Rl (cons r r1) 0 = a ->
pos_Rl (cons r r1)
  (Init.Nat.pred (Rlength (cons r r1))) = b ->
Int_SF (app_Rlist (mid_Rlist r1 r) f)
  (cons r r1) <=
Int_SF (app_Rlist (mid_Rlist r1 r) g)
  (cons r r1)
x:R
H4:r0 < x < b
H5:r0 < x
H6:x < b
pos_Rl (cons r (cons r0 r1)) 0 <= r0
  simpl; apply (H0 0%nat); simpl; apply lt_O_Sn.
Qed.

Lemma StepFun_P36 :
  forall (a b:R) (f g:StepFun a b) (l:Rlist),
    a <= b ->
    is_subdivision f a b l ->
    is_subdivision g a b l ->
    (forall x:R, a < x < b -> f x <= g x) ->
    RiemannInt_SF f <= RiemannInt_SF g.forall (a b : R) (f g : StepFun a b) (l : Rlist),
a <= b ->
is_subdivision f a b l ->
is_subdivision g a b l ->
(forall x : R, a < x < b -> f x <= g x) ->
RiemannInt_SF f <= RiemannInt_SF g
Proof.forall (a b : R) (f g : StepFun a b) (l : Rlist),
a <= b ->
is_subdivision f a b l ->
is_subdivision g a b l ->
(forall x : R, a < x < b -> f x <= g x) ->
RiemannInt_SF f <= RiemannInt_SF g
  intros; unfold RiemannInt_SF; decide (Rle_dec a b) with H.a, b:R
f, g:StepFun a b
l:Rlist
X:is_subdivision f a b l
X0:is_subdivision g a b l
H0:forall x : R, a < x < b -> f x <= g x
H:a <= b
Int_SF (subdivision_val f) (subdivision f) <=
Int_SF (subdivision_val g) (subdivision g)
  replace (Int_SF (subdivision_val f) (subdivision f)) with (Int_SF (FF l f) l).a, b:R
f, g:StepFun a b
l:Rlist
X:is_subdivision f a b l
X0:is_subdivision g a b l
H0:forall x : R, a < x < b -> f x <= g x
H:a <= b
Int_SF (FF l f) l <=
Int_SF (subdivision_val g) (subdivision g)
a, b:R
f, g:StepFun a b
l:Rlist
X:is_subdivision f a b l
X0:is_subdivision g a b l
H0:forall x : R, a < x < b -> f x <= g x
H:a <= b
Int_SF (FF l f) l =
Int_SF (subdivision_val f) (subdivision f)
  replace (Int_SF (subdivision_val g) (subdivision g)) with (Int_SF (FF l g) l).a, b:R
f, g:StepFun a b
l:Rlist
X:is_subdivision f a b l
X0:is_subdivision g a b l
H0:forall x : R, a < x < b -> f x <= g x
H:a <= b
Int_SF (FF l f) l <= Int_SF (FF l g) l
a, b:R
f, g:StepFun a b
l:Rlist
X:is_subdivision f a b l
X0:is_subdivision g a b l
H0:forall x : R, a < x < b -> f x <= g x
H:a <= b
Int_SF (FF l g) l =
Int_SF (subdivision_val g) (subdivision g)
a, b:R
f, g:StepFun a b
l:Rlist
X:is_subdivision f a b l
X0:is_subdivision g a b l
H0:forall x : R, a < x < b -> f x <= g x
H:a <= b
Int_SF (FF l f) l =
Int_SF (subdivision_val f) (subdivision f)
  unfold is_subdivision in X; elim X; clear X; intros;
    unfold adapted_couple in p; decompose [and] p; clear p;
      assert (H5 : Rmin a b = a);
        [ unfold Rmin; decide (Rle_dec a b) with H; reflexivity
          | assert (H7 : Rmax a b = b);
            [ unfold Rmax; decide (Rle_dec a b) with H; reflexivity
              | rewrite H5 in H3; rewrite H7 in H2; eapply StepFun_P35 with a b;
                assumption ] ].a, b:R
f, g:StepFun a b
l:Rlist
X:is_subdivision f a b l
X0:is_subdivision g a b l
H0:forall x : R, a < x < b -> f x <= g x
H:a <= b
Int_SF (FF l g) l =
Int_SF (subdivision_val g) (subdivision g)
a, b:R
f, g:StepFun a b
l:Rlist
X:is_subdivision f a b l
X0:is_subdivision g a b l
H0:forall x : R, a < x < b -> f x <= g x
H:a <= b
Int_SF (FF l f) l =
Int_SF (subdivision_val f) (subdivision f)
  apply StepFun_P17 with (fe g) a b;
    [ apply StepFun_P21; assumption | apply StepFun_P1 ].a, b:R
f, g:StepFun a b
l:Rlist
X:is_subdivision f a b l
X0:is_subdivision g a b l
H0:forall x : R, a < x < b -> f x <= g x
H:a <= b
Int_SF (FF l f) l =
Int_SF (subdivision_val f) (subdivision f)
  apply StepFun_P17 with (fe f) a b;
    [ apply StepFun_P21; assumption | apply StepFun_P1 ].
Qed.

Lemma StepFun_P37 :
  forall (a b:R) (f g:StepFun a b),
    a <= b ->
    (forall x:R, a < x < b -> f x <= g x) ->
    RiemannInt_SF f <= RiemannInt_SF g.forall (a b : R) (f g : StepFun a b),
a <= b ->
(forall x : R, a < x < b -> f x <= g x) ->
RiemannInt_SF f <= RiemannInt_SF g
Proof.forall (a b : R) (f g : StepFun a b),
a <= b ->
(forall x : R, a < x < b -> f x <= g x) ->
RiemannInt_SF f <= RiemannInt_SF g
  intros; eapply StepFun_P36; try assumption.a, b:R
f, g:StepFun a b
H:a <= b
H0:forall x : R, a < x < b -> f x <= g x
is_subdivision f a b ?l
a, b:R
f, g:StepFun a b
H:a <= b
H0:forall x : R, a < x < b -> f x <= g x
is_subdivision g a b ?l
  eapply StepFun_P25; apply StepFun_P29.a, b:R
f, g:StepFun a b
H:a <= b
H0:forall x : R, a < x < b -> f x <= g x
is_subdivision g a b
  (cons_ORlist (subdivision ?f) (subdivision f))
  eapply StepFun_P23; apply StepFun_P29.
Qed.

Lemma StepFun_P38 :
  forall (l:Rlist) (a b:R) (f:R -> R),
    ordered_Rlist l ->
    pos_Rl l 0 = a ->
    pos_Rl l (pred (Rlength l)) = b ->
    { g:StepFun a b |
      g b = f b /\
      (forall i:nat,
        (i < pred (Rlength l))%nat ->
        constant_D_eq g (co_interval (pos_Rl l i) (pos_Rl l (S i)))
        (f (pos_Rl l i))) }.forall (l : Rlist) (a b : R) (f : R -> R),
ordered_Rlist l ->
pos_Rl l 0 = a ->
pos_Rl l (Init.Nat.pred (Rlength l)) = b ->
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl l i) (pos_Rl l (S i)))
   (f (pos_Rl l i)))}
Proof.forall (l : Rlist) (a b : R) (f : R -> R),
ordered_Rlist l ->
pos_Rl l 0 = a ->
pos_Rl l (Init.Nat.pred (Rlength l)) = b ->
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength l))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl l i) (pos_Rl l (S i)))
   (f (pos_Rl l i)))}
  intros l a b f; generalize a; clear a; induction l.b:R
f:R -> R
forall a : R,
ordered_Rlist nil ->
pos_Rl nil 0 = a ->
pos_Rl nil (Init.Nat.pred (Rlength nil)) = b ->
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength nil))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl nil i) (pos_Rl nil (S i)))
   (f (pos_Rl nil i)))}
r:R
l:Rlist
b:R
f:R -> R
IHl:forall a : R,
ordered_Rlist l ->
pos_Rl l 0 = a ->
pos_Rl l (Init.Nat.pred (Rlength l)) = b ->
{g : StepFun a b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq g (co_interval (pos_Rl l i) (pos_Rl l (S i))) (f (pos_Rl l i)))}
forall a : R,
ordered_Rlist (cons r l) ->
pos_Rl (cons r l) 0 = a ->
pos_Rl (cons r l) (Init.Nat.pred (Rlength (cons r l))) =
b ->
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r l)))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r l) i)
      (pos_Rl (cons r l) (S i)))
   (f (pos_Rl (cons r l) i)))}
  intros a H H0 H1; simpl in H0; simpl in H1;
    exists (mkStepFun (StepFun_P4 a b (f b))); split.b:R
f:R -> R
a:R
H:ordered_Rlist nil
H0:0 = a
H1:0 = b
{| fe := fct_cte (f b); pre := StepFun_P4 a b (f b) |}
  b = f b
b:R
f:R -> R
a:R
H:ordered_Rlist nil
H0:0 = a
H1:0 = b
forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq
  {|
  fe := fct_cte (f b);
  pre := StepFun_P4 a b (f b) |}
  (co_interval (pos_Rl nil i) (pos_Rl nil (S i)))
  (f (pos_Rl nil i))
r:R
l:Rlist
b:R
f:R -> R
IHl:forall a : R,
ordered_Rlist l ->
pos_Rl l 0 = a ->
pos_Rl l (Init.Nat.pred (Rlength l)) = b ->
{g : StepFun a b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq g (co_interval (pos_Rl l i) (pos_Rl l (S i))) (f (pos_Rl l i)))}
forall a : R,
ordered_Rlist (cons r l) ->
pos_Rl (cons r l) 0 = a ->
pos_Rl (cons r l) (Init.Nat.pred (Rlength (cons r l))) =
b ->
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r l)))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r l) i)
      (pos_Rl (cons r l) (S i)))
   (f (pos_Rl (cons r l) i)))}
  reflexivity.b:R
f:R -> R
a:R
H:ordered_Rlist nil
H0:0 = a
H1:0 = b
forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq
  {|
  fe := fct_cte (f b);
  pre := StepFun_P4 a b (f b) |}
  (co_interval (pos_Rl nil i) (pos_Rl nil (S i)))
  (f (pos_Rl nil i))
r:R
l:Rlist
b:R
f:R -> R
IHl:forall a : R,
ordered_Rlist l ->
pos_Rl l 0 = a ->
pos_Rl l (Init.Nat.pred (Rlength l)) = b ->
{g : StepFun a b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq g (co_interval (pos_Rl l i) (pos_Rl l (S i))) (f (pos_Rl l i)))}
forall a : R,
ordered_Rlist (cons r l) ->
pos_Rl (cons r l) 0 = a ->
pos_Rl (cons r l) (Init.Nat.pred (Rlength (cons r l))) =
b ->
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r l)))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r l) i)
      (pos_Rl (cons r l) (S i)))
   (f (pos_Rl (cons r l) i)))}
  intros; elim (lt_n_O _ H2).r:R
l:Rlist
b:R
f:R -> R
IHl:forall a : R,
ordered_Rlist l ->
pos_Rl l 0 = a ->
pos_Rl l (Init.Nat.pred (Rlength l)) = b ->
{g : StepFun a b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength l))%nat ->
constant_D_eq g (co_interval (pos_Rl l i) (pos_Rl l (S i))) (f (pos_Rl l i)))}
forall a : R,
ordered_Rlist (cons r l) ->
pos_Rl (cons r l) 0 = a ->
pos_Rl (cons r l) (Init.Nat.pred (Rlength (cons r l))) =
b ->
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r l)))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r l) i)
      (pos_Rl (cons r l) (S i)))
   (f (pos_Rl (cons r l) i)))}
  intros; destruct l as [| r1 l].r, b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist nil ->
pos_Rl nil 0 = a0 ->
pos_Rl nil (Init.Nat.pred (Rlength nil)) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq g (co_interval (pos_Rl nil i) (pos_Rl nil (S i)))
(f (pos_Rl nil i)))}
a:R
H:ordered_Rlist (cons r nil)
H0:pos_Rl (cons r nil) 0 = a
H1:pos_Rl (cons r nil)
  (Init.Nat.pred (Rlength (cons r nil))) = b
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r nil) i)
      (pos_Rl (cons r nil) (S i)))
   (f (pos_Rl (cons r nil) i)))}
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  simpl in H1; simpl in H0; exists (mkStepFun (StepFun_P4 a b (f b))); split.r, b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist nil ->
pos_Rl nil 0 = a0 ->
pos_Rl nil (Init.Nat.pred (Rlength nil)) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq g (co_interval (pos_Rl nil i) (pos_Rl nil (S i)))
(f (pos_Rl nil i)))}
a:R
H:ordered_Rlist (cons r nil)
H0:r = a
H1:r = b
{| fe := fct_cte (f b); pre := StepFun_P4 a b (f b) |}
  b = f b
r, b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist nil ->
pos_Rl nil 0 = a0 ->
pos_Rl nil (Init.Nat.pred (Rlength nil)) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq g (co_interval (pos_Rl nil i) (pos_Rl nil (S i)))
(f (pos_Rl nil i)))}
a:R
H:ordered_Rlist (cons r nil)
H0:r = a
H1:r = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq
  {|
  fe := fct_cte (f b);
  pre := StepFun_P4 a b (f b) |}
  (co_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i)))
  (f (pos_Rl (cons r nil) i))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  reflexivity.r, b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist nil ->
pos_Rl nil 0 = a0 ->
pos_Rl nil (Init.Nat.pred (Rlength nil)) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq g (co_interval (pos_Rl nil i) (pos_Rl nil (S i)))
(f (pos_Rl nil i)))}
a:R
H:ordered_Rlist (cons r nil)
H0:r = a
H1:r = b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq
  {|
  fe := fct_cte (f b);
  pre := StepFun_P4 a b (f b) |}
  (co_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i)))
  (f (pos_Rl (cons r nil) i))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  intros i H2; elim (lt_n_O _ H2).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  intros; assert (H2 : ordered_Rlist (cons r1 l)).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
ordered_Rlist (cons r1 l)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  apply RList_P4 with r; assumption.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  assert (H3 : pos_Rl (cons r1 l) 0 = r1).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
pos_Rl (cons r1 l) 0 = r1
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  assert (H4 : pos_Rl (cons r1 l) (pred (Rlength (cons r1 l))) = b).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  rewrite <- H1; reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g : StepFun a0 b |
g b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
{g : StepFun a b |
g b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  elim (IHl r1 H2 H3 H4); intros g [H5 H6].r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  set
    (g' :=
      fun x:R => match Rle_dec r1 x with
                   | left _ => g x
                   | right _ => f a
                 end).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  assert (H7 : r1 <= b).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
r1 <= b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  rewrite <- H4; apply RList_P7; [ assumption | left; reflexivity ].r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  assert (H8 : IsStepFun g' a b).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
IsStepFun g' a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  unfold IsStepFun; assert (H8 := pre g); unfold IsStepFun in H8;
    elim H8; intros lg H9; unfold is_subdivision in H9;
      elim H9; clear H9; intros lg2 H9; split with (cons a lg);
        unfold is_subdivision; split with (cons (f a) lg2);
          unfold adapted_couple in H9; decompose [and] H9; clear H9;
            unfold adapted_couple; repeat split.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) = Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
(pos_Rl lg2 i)
ordered_Rlist (cons a lg)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) = Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
(pos_Rl lg2 i)
pos_Rl (cons a lg) 0 = Rmin a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) = Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
(pos_Rl lg2 i)
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) = Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
(pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) = Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
(pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  unfold ordered_Rlist; intros; simpl in H9;
    induction  i as [| i Hreci].r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
(f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) = Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
(pos_Rl lg2 i)
H9:(0 < Rlength lg)%nat
pos_Rl (cons a lg) 0 <= pos_Rl (cons a lg) 1
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l) (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g0
(co_interval (pos_Rl (cons r1 l) i0) (pos_Rl (cons r1 l) (S i0)))
(f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) = Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g (open_interval (pos_Rl lg i0) (pos_Rl lg (S i0)))
(pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
pos_Rl (cons a lg) i <=
pos_Rl (cons a lg) (S i)
pos_Rl (cons a lg) (S i) <=
pos_Rl (cons a lg) (S (S i))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg) 0 = Rmin a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  simpl; rewrite H12; replace (Rmin r1 b) with r1.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H9:(0 < Rlength lg)%nat
a <= r1
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H9:(0 < Rlength lg)%nat
r1 = Rmin r1 b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
pos_Rl (cons a lg) i <=
pos_Rl (cons a lg) (S i)
pos_Rl (cons a lg) (S i) <=
pos_Rl (cons a lg) (S (S i))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg) 0 = Rmin a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  simpl in H0; rewrite <- H0; apply (H 0%nat); simpl; apply lt_O_Sn.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H9:(0 < Rlength lg)%nat
r1 = Rmin r1 b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
pos_Rl (cons a lg) i <=
pos_Rl (cons a lg) (S i)
pos_Rl (cons a lg) (S i) <=
pos_Rl (cons a lg) (S (S i))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg) 0 = Rmin a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  unfold Rmin; decide (Rle_dec r1 b) with H7; reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
pos_Rl (cons a lg) i <=
pos_Rl (cons a lg) (S i)
pos_Rl (cons a lg) (S i) <=
pos_Rl (cons a lg) (S (S i))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg) 0 = Rmin a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  apply (H10 i); apply lt_S_n.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
pos_Rl (cons a lg) i <=
pos_Rl (cons a lg) (S i)
(S i < S (Init.Nat.pred (Rlength lg)))%nat
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg) 0 = Rmin a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  replace (S (pred (Rlength lg))) with (Rlength lg).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
pos_Rl (cons a lg) i <=
pos_Rl (cons a lg) (S i)
(S i < Rlength lg)%nat
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
pos_Rl (cons a lg) i <=
pos_Rl (cons a lg) (S i)
Rlength lg = S (Init.Nat.pred (Rlength lg))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg) 0 = Rmin a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  apply H9.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
pos_Rl (cons a lg) i <=
pos_Rl (cons a lg) (S i)
Rlength lg = S (Init.Nat.pred (Rlength lg))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg) 0 = Rmin a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  apply S_pred with 0%nat; apply neq_O_lt; intro; rewrite <- H14 in H9;
    elim (lt_n_O _ H9).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg) 0 = Rmin a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  simpl; assert (H14 : a <= b).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
a <= b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H14:a <= b
a = Rmin a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  rewrite <- H1; simpl in H0; rewrite <- H0; apply RList_P7;
    [ assumption | left; reflexivity ].r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H14:a <= b
a = Rmin a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  unfold Rmin; decide (Rle_dec a b) with H14; reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  assert (H14 : a <= b).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
a <= b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H14:a <= b
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  rewrite <- H1; simpl in H0; rewrite <- H0; apply RList_P7;
    [ assumption | left; reflexivity ].r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H14:a <= b
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  replace (Rmax a b) with (Rmax r1 b).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H14:a <= b
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) = 
Rmax r1 b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H14:a <= b
Rmax r1 b = Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  rewrite <- H11; induction  lg as [| r0 lg Hreclg].r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg2:Rlist
H10:ordered_Rlist nil
H12:pos_Rl nil 0 = Rmin r1 b
H11:pos_Rl nil (Init.Nat.pred (Rlength nil)) =
Rmax r1 b
H13:Rlength nil = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq g
  (open_interval (pos_Rl nil i)
     (pos_Rl nil (S i))) 
  (pos_Rl lg2 i)
H14:a <= b
pos_Rl (cons a nil)
  (Init.Nat.pred (Rlength (cons a nil))) =
pos_Rl nil (Init.Nat.pred (Rlength nil))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
r0:R
lg, lg2:Rlist
H10:ordered_Rlist (cons r0 lg)
H12:pos_Rl (cons r0 lg) 0 = Rmin r1 b
H11:pos_Rl (cons r0 lg)
  (Init.Nat.pred (Rlength (cons r0 lg))) =
Rmax r1 b
H13:Rlength (cons r0 lg) = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r0 lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons r0 lg) i)
     (pos_Rl (cons r0 lg) (S i))) 
  (pos_Rl lg2 i)
H14:a <= b
Hreclg:ordered_Rlist lg ->
pos_Rl lg 0 = Rmin r1 b ->
pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b ->
Rlength lg = S (Rlength lg2) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lg))%nat ->
 constant_D_eq g
   (open_interval (pos_Rl lg i)
      (pos_Rl lg (S i))) 
   (pos_Rl lg2 i)) ->
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) =
pos_Rl lg (Init.Nat.pred (Rlength lg))
pos_Rl (cons a (cons r0 lg))
  (Init.Nat.pred (Rlength (cons a (cons r0 lg)))) =
pos_Rl (cons r0 lg)
  (Init.Nat.pred (Rlength (cons r0 lg)))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H14:a <= b
Rmax r1 b = Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  simpl in H13; discriminate.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
r0:R
lg, lg2:Rlist
H10:ordered_Rlist (cons r0 lg)
H12:pos_Rl (cons r0 lg) 0 = Rmin r1 b
H11:pos_Rl (cons r0 lg)
  (Init.Nat.pred (Rlength (cons r0 lg))) =
Rmax r1 b
H13:Rlength (cons r0 lg) = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r0 lg)))%nat ->
constant_D_eq g
  (open_interval (pos_Rl (cons r0 lg) i)
     (pos_Rl (cons r0 lg) (S i))) 
  (pos_Rl lg2 i)
H14:a <= b
Hreclg:ordered_Rlist lg ->
pos_Rl lg 0 = Rmin r1 b ->
pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b ->
Rlength lg = S (Rlength lg2) ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength lg))%nat ->
 constant_D_eq g
   (open_interval (pos_Rl lg i)
      (pos_Rl lg (S i))) 
   (pos_Rl lg2 i)) ->
pos_Rl (cons a lg)
  (Init.Nat.pred (Rlength (cons a lg))) =
pos_Rl lg (Init.Nat.pred (Rlength lg))
pos_Rl (cons a (cons r0 lg))
  (Init.Nat.pred (Rlength (cons a (cons r0 lg)))) =
pos_Rl (cons r0 lg)
  (Init.Nat.pred (Rlength (cons r0 lg)))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H14:a <= b
Rmax r1 b = Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H14:a <= b
Rmax r1 b = Rmax a b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  unfold Rmax; decide (Rle_dec a b) with H14; decide (Rle_dec r1 b) with H7;
    reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength (cons a lg) = S (Rlength (cons (f a) lg2))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  simpl; rewrite H13; reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons a lg)))%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  intros; simpl in H9; induction  i as [| i Hreci].r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H9:(0 < Rlength lg)%nat
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) 0)
     (pos_Rl (cons a lg) 1))
  (pos_Rl (cons (f a) lg2) 0)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) (S i))
     (pos_Rl (cons a lg) (S (S i))))
  (pos_Rl (cons (f a) lg2) (S i))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  unfold constant_D_eq, open_interval; simpl; intros;
    assert (H16 : Rmin r1 b = r1).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H9:(0 < Rlength lg)%nat
x:R
H14:a < x < pos_Rl lg 0
Rmin r1 b = r1
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H9:(0 < Rlength lg)%nat
x:R
H14:a < x < pos_Rl lg 0
H16:Rmin r1 b = r1
g' x = f a
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) (S i))
     (pos_Rl (cons a lg) (S (S i))))
  (pos_Rl (cons (f a) lg2) (S i))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  unfold Rmin; decide (Rle_dec r1 b) with H7; reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H9:(0 < Rlength lg)%nat
x:R
H14:a < x < pos_Rl lg 0
H16:Rmin r1 b = r1
g' x = f a
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) (S i))
     (pos_Rl (cons a lg) (S (S i))))
  (pos_Rl (cons (f a) lg2) (S i))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  rewrite H16 in H12; rewrite H12 in H14; elim H14; clear H14; intros _ H14;
    unfold g'; case (Rle_dec r1 x); intro r3.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = r1
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H9:(0 < Rlength lg)%nat
x:R
H16:Rmin r1 b = r1
H14:x < r1
r3:r1 <= x
g x = f a
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = r1
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H9:(0 < Rlength lg)%nat
x:R
H16:Rmin r1 b = r1
H14:x < r1
r3:~ r1 <= x
f a = f a
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) (S i))
     (pos_Rl (cons a lg) (S (S i))))
  (pos_Rl (cons (f a) lg2) (S i))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H14)).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = r1
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i : nat,
(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H9:(0 < Rlength lg)%nat
x:R
H16:Rmin r1 b = r1
H14:x < r1
r3:~ r1 <= x
f a = f a
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) (S i))
     (pos_Rl (cons a lg) (S (S i))))
  (pos_Rl (cons (f a) lg2) (S i))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
Hreci:(i < Rlength lg)%nat ->
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) i)
     (pos_Rl (cons a lg) (S i)))
  (pos_Rl (cons (f a) lg2) i)
constant_D_eq g'
  (open_interval (pos_Rl (cons a lg) (S i))
     (pos_Rl (cons a lg) (S (S i))))
  (pos_Rl (cons (f a) lg2) (S i))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  change
    (constant_D_eq g' (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
      (pos_Rl lg2 i)); clear Hreci; assert (H16 := H15 i);
    assert (H17 : (i < pred (Rlength lg))%nat).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
(i < Init.Nat.pred (Rlength lg))%nat
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
constant_D_eq g'
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  apply lt_S_n.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
(S i < S (Init.Nat.pred (Rlength lg)))%nat
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
constant_D_eq g'
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  replace (S (pred (Rlength lg))) with (Rlength lg).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
(S i < Rlength lg)%nat
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength lg = S (Init.Nat.pred (Rlength lg))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
constant_D_eq g'
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  assumption.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
Rlength lg = S (Init.Nat.pred (Rlength lg))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
constant_D_eq g'
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  apply S_pred with 0%nat; apply neq_O_lt; red; intro;
    rewrite <- H14 in H9; elim (lt_n_O _ H9).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
constant_D_eq g'
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  assert (H18 := H16 H17); unfold constant_D_eq, open_interval in H18;
    unfold constant_D_eq, open_interval; intros;
      assert (H19 := H18 _ H14); rewrite <- H19; unfold g';
        case (Rle_dec r1 x) as [|[]].r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H14:pos_Rl lg i < x < pos_Rl lg (S i)
H19:g x = pos_Rl lg2 i
r0:r1 <= x
g x = g x
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H14:pos_Rl lg i < x < pos_Rl lg (S i)
H19:g x = pos_Rl lg2 i
r1 <= x
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H14:pos_Rl lg i < x < pos_Rl lg (S i)
H19:g x = pos_Rl lg2 i
r1 <= x
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  replace r1 with (Rmin r1 b).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H14:pos_Rl lg i < x < pos_Rl lg (S i)
H19:g x = pos_Rl lg2 i
Rmin r1 b <= x
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H14:pos_Rl lg i < x < pos_Rl lg (S i)
H19:g x = pos_Rl lg2 i
Rmin r1 b = r1
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  rewrite <- H12; elim H14; clear H14; intros H14 _; left;
    apply Rle_lt_trans with (pos_Rl lg i); try assumption.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H19:g x = pos_Rl lg2 i
H14:pos_Rl lg i < x
pos_Rl lg 0 <= pos_Rl lg i
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H14:pos_Rl lg i < x < pos_Rl lg (S i)
H19:g x = pos_Rl lg2 i
Rmin r1 b = r1
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  apply RList_P5.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H19:g x = pos_Rl lg2 i
H14:pos_Rl lg i < x
ordered_Rlist lg
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H19:g x = pos_Rl lg2 i
H14:pos_Rl lg i < x
In (pos_Rl lg i) lg
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H14:pos_Rl lg i < x < pos_Rl lg (S i)
H19:g x = pos_Rl lg2 i
Rmin r1 b = r1
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  assumption.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H19:g x = pos_Rl lg2 i
H14:pos_Rl lg i < x
In (pos_Rl lg i) lg
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H14:pos_Rl lg i < x < pos_Rl lg (S i)
H19:g x = pos_Rl lg2 i
Rmin r1 b = r1
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  elim (RList_P3 lg (pos_Rl lg i)); intros; apply H21; exists i; split.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H19:g x = pos_Rl lg2 i
H14:pos_Rl lg i < x
H20:In (pos_Rl lg i) lg ->
exists i0 : nat,
  pos_Rl lg i = pos_Rl lg i0 /\
  (i0 < Rlength lg)%nat
H21:(exists i0 : nat,
   pos_Rl lg i = pos_Rl lg i0 /\
   (i0 < Rlength lg)%nat) -> 
In (pos_Rl lg i) lg
pos_Rl lg i = pos_Rl lg i
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H19:g x = pos_Rl lg2 i
H14:pos_Rl lg i < x
H20:In (pos_Rl lg i) lg ->
exists i0 : nat,
  pos_Rl lg i = pos_Rl lg i0 /\
  (i0 < Rlength lg)%nat
H21:(exists i0 : nat,
   pos_Rl lg i = pos_Rl lg i0 /\
   (i0 < Rlength lg)%nat) -> 
In (pos_Rl lg i) lg
(i < Rlength lg)%nat
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H14:pos_Rl lg i < x < pos_Rl lg (S i)
H19:g x = pos_Rl lg2 i
Rmin r1 b = r1
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H19:g x = pos_Rl lg2 i
H14:pos_Rl lg i < x
H20:In (pos_Rl lg i) lg ->
exists i0 : nat,
  pos_Rl lg i = pos_Rl lg i0 /\
  (i0 < Rlength lg)%nat
H21:(exists i0 : nat,
   pos_Rl lg i = pos_Rl lg i0 /\
   (i0 < Rlength lg)%nat) -> 
In (pos_Rl lg i) lg
(i < Rlength lg)%nat
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H14:pos_Rl lg i < x < pos_Rl lg (S i)
H19:g x = pos_Rl lg2 i
Rmin r1 b = r1
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  apply lt_trans with (pred (Rlength lg)); try assumption.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H19:g x = pos_Rl lg2 i
H14:pos_Rl lg i < x
H20:In (pos_Rl lg i) lg ->
exists i0 : nat,
  pos_Rl lg i = pos_Rl lg i0 /\
  (i0 < Rlength lg)%nat
H21:(exists i0 : nat,
   pos_Rl lg i = pos_Rl lg i0 /\
   (i0 < Rlength lg)%nat) -> 
In (pos_Rl lg i) lg
(Init.Nat.pred (Rlength lg) < Rlength lg)%nat
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H14:pos_Rl lg i < x < pos_Rl lg (S i)
H19:g x = pos_Rl lg2 i
Rmin r1 b = r1
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H22 in H17;
    elim (lt_n_O _ H17).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:{l0 : Rlist & is_subdivision g r1 b l0}
lg, lg2:Rlist
H10:ordered_Rlist lg
H12:pos_Rl lg 0 = Rmin r1 b
H11:pos_Rl lg (Init.Nat.pred (Rlength lg)) =
Rmax r1 b
H13:Rlength lg = S (Rlength lg2)
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i0)
     (pos_Rl lg (S i0))) 
  (pos_Rl lg2 i0)
i:nat
H9:(S i < Rlength lg)%nat
H16:(i < Init.Nat.pred (Rlength lg))%nat ->
constant_D_eq g
  (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
  (pos_Rl lg2 i)
H17:(i < Init.Nat.pred (Rlength lg))%nat
H18:forall x0 : R,
pos_Rl lg i < x0 < pos_Rl lg (S i) ->
g x0 = pos_Rl lg2 i
x:R
H14:pos_Rl lg i < x < pos_Rl lg (S i)
H19:g x = pos_Rl lg2 i
Rmin r1 b = r1
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  unfold Rmin; decide (Rle_dec r1 b) with H7; reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{g0 : StepFun a b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r (cons r1 l)) i)
      (pos_Rl (cons r (cons r1 l)) (S i)))
   (f (pos_Rl (cons r (cons r1 l)) i)))}
  exists (mkStepFun H8); split.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
{| fe := g'; pre := H8 |} b = f b
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
constant_D_eq {| fe := g'; pre := H8 |}
  (co_interval (pos_Rl (cons r (cons r1 l)) i)
     (pos_Rl (cons r (cons r1 l)) (S i)))
  (f (pos_Rl (cons r (cons r1 l)) i))
  simpl; unfold g'; decide (Rle_dec r1 b) with H7; assumption.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r1 l))))%nat ->
constant_D_eq {| fe := g'; pre := H8 |}
  (co_interval (pos_Rl (cons r (cons r1 l)) i)
     (pos_Rl (cons r (cons r1 l)) (S i)))
  (f (pos_Rl (cons r (cons r1 l)) i))
  intros; simpl in H9; induction  i as [| i Hreci].r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
H9:(0 < S (Rlength l))%nat
constant_D_eq {| fe := g'; pre := H8 |}
  (co_interval (pos_Rl (cons r (cons r1 l)) 0)
     (pos_Rl (cons r (cons r1 l)) 1))
  (f (pos_Rl (cons r (cons r1 l)) 0))
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
i:nat
H9:(S i < S (Rlength l))%nat
Hreci:(i < S (Rlength l))%nat ->
constant_D_eq {| fe := g'; pre := H8 |}
  (co_interval (pos_Rl (cons r (cons r1 l)) i)
     (pos_Rl (cons r (cons r1 l)) (S i)))
  (f (pos_Rl (cons r (cons r1 l)) i))
constant_D_eq {| fe := g'; pre := H8 |}
  (co_interval (pos_Rl (cons r (cons r1 l)) (S i))
     (pos_Rl (cons r (cons r1 l)) (S (S i))))
  (f (pos_Rl (cons r (cons r1 l)) (S i)))
  unfold constant_D_eq, co_interval; simpl; intros; simpl in H0;
    rewrite H0; elim H10; clear H10; intros; unfold g';
      case (Rle_dec r1 x); intro r3.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:r = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
H9:(0 < S (Rlength l))%nat
x:R
H10:r <= x
H11:x < r1
r3:r1 <= x
g x = f a
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:r = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
H9:(0 < S (Rlength l))%nat
x:R
H10:r <= x
H11:x < r1
r3:~ r1 <= x
f a = f a
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
i:nat
H9:(S i < S (Rlength l))%nat
Hreci:(i < S (Rlength l))%nat ->
constant_D_eq {| fe := g'; pre := H8 |}
  (co_interval (pos_Rl (cons r (cons r1 l)) i)
     (pos_Rl (cons r (cons r1 l)) (S i)))
  (f (pos_Rl (cons r (cons r1 l)) i))
constant_D_eq {| fe := g'; pre := H8 |}
  (co_interval (pos_Rl (cons r (cons r1 l)) (S i))
     (pos_Rl (cons r (cons r1 l)) (S (S i))))
  (f (pos_Rl (cons r (cons r1 l)) (S i)))
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H11)).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i : nat,
 (i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i)
      (pos_Rl (cons r1 l) (S i)))
   (f (pos_Rl (cons r1 l) i)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:r = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
H9:(0 < S (Rlength l))%nat
x:R
H10:r <= x
H11:x < r1
r3:~ r1 <= x
f a = f a
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
i:nat
H9:(S i < S (Rlength l))%nat
Hreci:(i < S (Rlength l))%nat ->
constant_D_eq {| fe := g'; pre := H8 |}
  (co_interval (pos_Rl (cons r (cons r1 l)) i)
     (pos_Rl (cons r (cons r1 l)) (S i)))
  (f (pos_Rl (cons r (cons r1 l)) i))
constant_D_eq {| fe := g'; pre := H8 |}
  (co_interval (pos_Rl (cons r (cons r1 l)) (S i))
     (pos_Rl (cons r (cons r1 l)) (S (S i))))
  (f (pos_Rl (cons r (cons r1 l)) (S i)))
  reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
i:nat
H9:(S i < S (Rlength l))%nat
Hreci:(i < S (Rlength l))%nat ->
constant_D_eq {| fe := g'; pre := H8 |}
  (co_interval (pos_Rl (cons r (cons r1 l)) i)
     (pos_Rl (cons r (cons r1 l)) (S i)))
  (f (pos_Rl (cons r (cons r1 l)) i))
constant_D_eq {| fe := g'; pre := H8 |}
  (co_interval (pos_Rl (cons r (cons r1 l)) (S i))
     (pos_Rl (cons r (cons r1 l)) (S (S i))))
  (f (pos_Rl (cons r (cons r1 l)) (S i)))
  clear Hreci;
    change
      (constant_D_eq (mkStepFun H8)
        (co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
        (f (pos_Rl (cons r1 l) i))); assert (H10 := H6 i);
      assert (H11 : (i < pred (Rlength (cons r1 l)))%nat).r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
i:nat
H9:(S i < S (Rlength l))%nat
H10:(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
(i < Init.Nat.pred (Rlength (cons r1 l)))%nat
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
i:nat
H9:(S i < S (Rlength l))%nat
H10:(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
H11:(i < Init.Nat.pred (Rlength (cons r1 l)))%nat
constant_D_eq {| fe := g'; pre := H8 |}
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
  simpl; apply lt_S_n; assumption.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x : R => if Rle_dec r1 x then g x else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
i:nat
H9:(S i < S (Rlength l))%nat
H10:(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
H11:(i < Init.Nat.pred (Rlength (cons r1 l)))%nat
constant_D_eq {| fe := g'; pre := H8 |}
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
  assert (H12 := H10 H11); unfold constant_D_eq, co_interval in H12;
    unfold constant_D_eq, co_interval; intros;
      rewrite <- (H12 _ H13); simpl; unfold g';
        case (Rle_dec r1 x) as [|[]].r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
i:nat
H9:(S i < S (Rlength l))%nat
H10:(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
H11:(i < Init.Nat.pred (Rlength (cons r1 l)))%nat
H12:forall x0 : R,
pos_Rl (cons r1 l) i <= x0 <
pos_Rl (cons r1 l) (S i) ->
g x0 = f (pos_Rl (cons r1 l) i)
x:R
H13:pos_Rl (cons r1 l) i <= x <
pos_Rl (cons r1 l) (S i)
r0:r1 <= x
g x = g x
r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
i:nat
H9:(S i < S (Rlength l))%nat
H10:(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
H11:(i < Init.Nat.pred (Rlength (cons r1 l)))%nat
H12:forall x0 : R,
pos_Rl (cons r1 l) i <= x0 <
pos_Rl (cons r1 l) (S i) ->
g x0 = f (pos_Rl (cons r1 l) i)
x:R
H13:pos_Rl (cons r1 l) i <= x <
pos_Rl (cons r1 l) (S i)
r1 <= x
  reflexivity.r, r1:R
l:Rlist
b:R
f:R -> R
IHl:forall a0 : R,
ordered_Rlist (cons r1 l) ->
pos_Rl (cons r1 l) 0 = a0 ->
pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b ->
{g0 : StepFun a0 b |
g0 b = f b /\
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
 constant_D_eq g0
   (co_interval (pos_Rl (cons r1 l) i0)
      (pos_Rl (cons r1 l) (S i0)))
   (f (pos_Rl (cons r1 l) i0)))}
a:R
H:ordered_Rlist (cons r (cons r1 l))
H0:pos_Rl (cons r (cons r1 l)) 0 = a
H1:pos_Rl (cons r (cons r1 l))
  (Init.Nat.pred (Rlength (cons r (cons r1 l)))) =
b
H2:ordered_Rlist (cons r1 l)
H3:pos_Rl (cons r1 l) 0 = r1
H4:pos_Rl (cons r1 l)
  (Init.Nat.pred (Rlength (cons r1 l))) = b
g:StepFun r1 b
H5:g b = f b
H6:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i0)
     (pos_Rl (cons r1 l) (S i0)))
  (f (pos_Rl (cons r1 l) i0))
g':=fun x0 : R => if Rle_dec r1 x0 then g x0 else f a:R -> R
H7:r1 <= b
H8:IsStepFun g' a b
i:nat
H9:(S i < S (Rlength l))%nat
H10:(i < Init.Nat.pred (Rlength (cons r1 l)))%nat ->
constant_D_eq g
  (co_interval (pos_Rl (cons r1 l) i)
     (pos_Rl (cons r1 l) (S i)))
  (f (pos_Rl (cons r1 l) i))
H11:(i < Init.Nat.pred (Rlength (cons r1 l)))%nat
H12:forall x0 : R,
pos_Rl (cons r1 l) i <= x0 <
pos_Rl (cons r1 l) (S i) ->
g x0 = f (pos_Rl (cons r1 l) i)
x:R
H13:pos_Rl (cons r1 l) i <= x <
pos_Rl (cons r1 l) (S i)
r1 <= x
  elim H13; clear H13; intros;
    apply Rle_trans with (pos_Rl (cons r1 l) i); try assumption;
      change (pos_Rl (cons r1 l) 0 <= pos_Rl (cons r1 l) i);
        elim (RList_P6 (cons r1 l)); intros; apply H15;
          [ assumption
            | apply le_O_n
            | simpl; apply lt_trans with (Rlength l);
              [ apply lt_S_n; assumption | apply lt_n_Sn ] ].
Qed.

Lemma StepFun_P39 :
  forall (a b:R) (f:StepFun a b),
    RiemannInt_SF f = - RiemannInt_SF (mkStepFun (StepFun_P6 (pre f))).forall (a b : R) (f : StepFun a b),
RiemannInt_SF f =
-
RiemannInt_SF {| fe := f; pre := StepFun_P6 (pre f) |}
Proof.forall (a b : R) (f : StepFun a b),
RiemannInt_SF f =
-
RiemannInt_SF {| fe := f; pre := StepFun_P6 (pre f) |}
  intros; unfold RiemannInt_SF; case (Rle_dec a b); case (Rle_dec b a);
    intros.a, b:R
f:StepFun a b
r:b <= a
r0:a <= b
Int_SF (subdivision_val f) (subdivision f) =
-
Int_SF
  (subdivision_val
     {| fe := f; pre := StepFun_P6 (pre f) |})
  (subdivision
     {| fe := f; pre := StepFun_P6 (pre f) |})
a, b:R
f:StepFun a b
n:~ b <= a
r:a <= b
Int_SF (subdivision_val f) (subdivision f) =
-
-
Int_SF
  (subdivision_val
     {| fe := f; pre := StepFun_P6 (pre f) |})
  (subdivision
     {| fe := f; pre := StepFun_P6 (pre f) |})
a, b:R
f:StepFun a b
r:b <= a
n:~ a <= b
- Int_SF (subdivision_val f) (subdivision f) =
-
Int_SF
  (subdivision_val
     {| fe := f; pre := StepFun_P6 (pre f) |})
  (subdivision
     {| fe := f; pre := StepFun_P6 (pre f) |})
a, b:R
f:StepFun a b
n:~ b <= a
n0:~ a <= b
- Int_SF (subdivision_val f) (subdivision f) =
-
-
Int_SF
  (subdivision_val
     {| fe := f; pre := StepFun_P6 (pre f) |})
  (subdivision
     {| fe := f; pre := StepFun_P6 (pre f) |})
  assert (H : adapted_couple f a b (subdivision f) (subdivision_val f));
    [ apply StepFun_P1
      | assert
        (H0 :
          adapted_couple (mkStepFun (StepFun_P6 (pre f))) b a
          (subdivision (mkStepFun (StepFun_P6 (pre f))))
          (subdivision_val (mkStepFun (StepFun_P6 (pre f)))));
        [ apply StepFun_P1
          | assert (H1 : a = b);
            [ apply Rle_antisym; assumption
              | rewrite (StepFun_P8 H H1); assert (H2 : b = a);
                [ symmetry ; apply H1 | rewrite (StepFun_P8 H0 H2); ring ] ] ] ].a, b:R
f:StepFun a b
n:~ b <= a
r:a <= b
Int_SF (subdivision_val f) (subdivision f) =
-
-
Int_SF
  (subdivision_val
     {| fe := f; pre := StepFun_P6 (pre f) |})
  (subdivision
     {| fe := f; pre := StepFun_P6 (pre f) |})
a, b:R
f:StepFun a b
r:b <= a
n:~ a <= b
- Int_SF (subdivision_val f) (subdivision f) =
-
Int_SF
  (subdivision_val
     {| fe := f; pre := StepFun_P6 (pre f) |})
  (subdivision
     {| fe := f; pre := StepFun_P6 (pre f) |})
a, b:R
f:StepFun a b
n:~ b <= a
n0:~ a <= b
- Int_SF (subdivision_val f) (subdivision f) =
-
-
Int_SF
  (subdivision_val
     {| fe := f; pre := StepFun_P6 (pre f) |})
  (subdivision
     {| fe := f; pre := StepFun_P6 (pre f) |})
  rewrite Ropp_involutive; eapply StepFun_P17;
    [ apply StepFun_P1
      | apply StepFun_P2; set (H := StepFun_P6 (pre f)); unfold IsStepFun in H;
        elim H; intros; unfold is_subdivision;
          elim p; intros; apply p0 ].a, b:R
f:StepFun a b
r:b <= a
n:~ a <= b
- Int_SF (subdivision_val f) (subdivision f) =
-
Int_SF
  (subdivision_val
     {| fe := f; pre := StepFun_P6 (pre f) |})
  (subdivision
     {| fe := f; pre := StepFun_P6 (pre f) |})
a, b:R
f:StepFun a b
n:~ b <= a
n0:~ a <= b
- Int_SF (subdivision_val f) (subdivision f) =
-
-
Int_SF
  (subdivision_val
     {| fe := f; pre := StepFun_P6 (pre f) |})
  (subdivision
     {| fe := f; pre := StepFun_P6 (pre f) |})
  apply Ropp_eq_compat; eapply StepFun_P17;
    [ apply StepFun_P1
      | apply StepFun_P2; set (H := StepFun_P6 (pre f)); unfold IsStepFun in H;
        elim H; intros; unfold is_subdivision;
          elim p; intros; apply p0 ].a, b:R
f:StepFun a b
n:~ b <= a
n0:~ a <= b
- Int_SF (subdivision_val f) (subdivision f) =
-
-
Int_SF
  (subdivision_val
     {| fe := f; pre := StepFun_P6 (pre f) |})
  (subdivision
     {| fe := f; pre := StepFun_P6 (pre f) |})
  assert (H : a < b);
    [ auto with real
      | assert (H0 : b < a);
        [ auto with real | elim (Rlt_irrefl _ (Rlt_trans _ _ _ H H0)) ] ].
Qed.

Lemma StepFun_P40 :
  forall (f:R -> R) (a b c:R) (l1 l2 lf1 lf2:Rlist),
    a < b ->
    b < c ->
    adapted_couple f a b l1 lf1 ->
    adapted_couple f b c l2 lf2 ->
    adapted_couple f a c (cons_Rlist l1 l2) (FF (cons_Rlist l1 l2) f).forall (f : R -> R) (a b c : R)
  (l1 l2 lf1 lf2 : Rlist),
a < b ->
b < c ->
adapted_couple f a b l1 lf1 ->
adapted_couple f b c l2 lf2 ->
adapted_couple f a c (cons_Rlist l1 l2)
  (FF (cons_Rlist l1 l2) f)
Proof.forall (f : R -> R) (a b c : R)
  (l1 l2 lf1 lf2 : Rlist),
a < b ->
b < c ->
adapted_couple f a b l1 lf1 ->
adapted_couple f b c l2 lf2 ->
adapted_couple f a c (cons_Rlist l1 l2)
  (FF (cons_Rlist l1 l2) f)
  intros f a b c l1 l2 lf1 lf2 H H0 H1 H2; unfold adapted_couple in H1, H2;
    unfold adapted_couple; decompose [and] H1;
      decompose [and] H2; clear H1 H2; repeat split.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
ordered_Rlist (cons_Rlist l1 l2)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
pos_Rl (cons_Rlist l1 l2) 0 = Rmin a c
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
pos_Rl (cons_Rlist l1 l2)
  (Init.Nat.pred (Rlength (cons_Rlist l1 l2))) =
Rmax a c
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
Rlength (cons_Rlist l1 l2) =
S (Rlength (FF (cons_Rlist l1 l2) f))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  apply RList_P25; try assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
pos_Rl l1 (Init.Nat.pred (Rlength l1)) <= pos_Rl l2 0
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
pos_Rl (cons_Rlist l1 l2) 0 = Rmin a c
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
pos_Rl (cons_Rlist l1 l2)
  (Init.Nat.pred (Rlength (cons_Rlist l1 l2))) =
Rmax a c
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
Rlength (cons_Rlist l1 l2) =
S (Rlength (FF (cons_Rlist l1 l2) f))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  rewrite H10; rewrite H4; unfold Rmin, Rmax; case (Rle_dec a b) as [|[]];
    case (Rle_dec b c) as [|[]];
      (right; reflexivity) || (left; assumption).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
pos_Rl (cons_Rlist l1 l2) 0 = Rmin a c
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
pos_Rl (cons_Rlist l1 l2)
  (Init.Nat.pred (Rlength (cons_Rlist l1 l2))) =
Rmax a c
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
Rlength (cons_Rlist l1 l2) =
S (Rlength (FF (cons_Rlist l1 l2) f))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  rewrite RList_P22.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
pos_Rl l1 0 = Rmin a c
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
l1 <> nil
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
pos_Rl (cons_Rlist l1 l2)
  (Init.Nat.pred (Rlength (cons_Rlist l1 l2))) =
Rmax a c
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
Rlength (cons_Rlist l1 l2) =
S (Rlength (FF (cons_Rlist l1 l2) f))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  rewrite H5; unfold Rmin, Rmax; case (Rle_dec a c) as [|[]]; case (Rle_dec a b) as [|[]];
      [ reflexivity
        | left; assumption
        | apply Rle_trans with b; left; assumption
        | left; assumption ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
l1 <> nil
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
pos_Rl (cons_Rlist l1 l2)
  (Init.Nat.pred (Rlength (cons_Rlist l1 l2))) =
Rmax a c
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
Rlength (cons_Rlist l1 l2) =
S (Rlength (FF (cons_Rlist l1 l2) f))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  red; intro; rewrite H1 in H6; discriminate.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
pos_Rl (cons_Rlist l1 l2)
  (Init.Nat.pred (Rlength (cons_Rlist l1 l2))) =
Rmax a c
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
Rlength (cons_Rlist l1 l2) =
S (Rlength (FF (cons_Rlist l1 l2) f))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  rewrite RList_P24.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax a c
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
l2 <> nil
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
Rlength (cons_Rlist l1 l2) =
S (Rlength (FF (cons_Rlist l1 l2) f))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  rewrite H9; unfold Rmin, Rmax; case (Rle_dec a c) as [|[]]; case (Rle_dec b c) as [|[]];
      [ reflexivity
        | left; assumption
        | apply Rle_trans with b; left; assumption
        | left; assumption ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
l2 <> nil
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
Rlength (cons_Rlist l1 l2) =
S (Rlength (FF (cons_Rlist l1 l2) f))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  red; intro; rewrite H1 in H11; discriminate.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
Rlength (cons_Rlist l1 l2) =
S (Rlength (FF (cons_Rlist l1 l2) f))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  apply StepFun_P20.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
(0 < Rlength (cons_Rlist l1 l2))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  rewrite RList_P23; apply neq_O_lt; red; intro.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
H1:0%nat = (Rlength l1 + Rlength l2)%nat
False
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  assert (H2 : (Rlength l1 + Rlength l2)%nat = 0%nat).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
H1:0%nat = (Rlength l1 + Rlength l2)%nat
(Rlength l1 + Rlength l2)%nat = 0%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
H1:0%nat = (Rlength l1 + Rlength l2)%nat
H2:(Rlength l1 + Rlength l2)%nat = 0%nat
False
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  symmetry ; apply H1.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
H1:0%nat = (Rlength l1 + Rlength l2)%nat
H2:(Rlength l1 + Rlength l2)%nat = 0%nat
False
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  elim (plus_is_O _ _ H2); intros; rewrite H12 in H6; discriminate.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons_Rlist l1 l2) i)
     (pos_Rl (cons_Rlist l1 l2) (S i)))
  (pos_Rl (FF (cons_Rlist l1 l2) f) i)
  unfold constant_D_eq, open_interval; intros;
    elim (le_or_lt (S (S i)) (Rlength l1)); intro.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(S (S i) <= Rlength l1)%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  assert (H14 : pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(S (S i) <= Rlength l1)%nat
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  apply RList_P26; apply lt_S_n; apply le_lt_n_Sm; apply le_S_n;
    apply le_trans with (Rlength l1); [ assumption | apply le_n_Sn ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  assert (H15 : pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l1 (S i)).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l1 (S i)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l1 (S i)
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  apply RList_P26; apply lt_S_n; apply le_lt_n_Sm; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l1 (S i)
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite H14 in H2; rewrite H15 in H2; assert (H16 : (2 <= Rlength l1)%nat).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l1 (S i)
(2 <= Rlength l1)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  apply le_trans with (S (S i));
    [ repeat apply le_n_S; apply le_O_n | assumption ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  elim (RList_P20 _ H16); intros r1 [r2 [r3 H17]]; rewrite H17;
    change
      (f x = pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i)
     ; rewrite RList_P12.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     i)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  induction  i as [| i Hreci].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
(pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     0)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f (open_interval (pos_Rl l2 i0) (pos_Rl l2 (S i0)))
(pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) = pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  simpl; assert (H18 := H8 0%nat);
    unfold constant_D_eq, open_interval in H18;
      assert (H19 : (0 < pred (Rlength l1))%nat).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
(0 < Init.Nat.pred (Rlength l1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
f x = f ((r1 + r2) / 2)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite H17; simpl; apply lt_O_Sn.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
f x = f ((r1 + r2) / 2)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  assert (H20 := H18 H19); repeat rewrite H20.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
pos_Rl lf1 0 = pos_Rl lf1 0
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
pos_Rl l1 0 < (r1 + r2) / 2 < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
pos_Rl l1 0 < x < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  reflexivity.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
pos_Rl l1 0 < (r1 + r2) / 2 < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
pos_Rl l1 0 < x < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  assert (H21 : r1 <= r2).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
r1 <= r2
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H21:r1 <= r2
pos_Rl l1 0 < (r1 + r2) / 2 < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
pos_Rl l1 0 < x < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite H17 in H3; apply (H3 0%nat).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
r1, r2:R
r3:Rlist
H3:ordered_Rlist (cons r1 (cons r2 r3))
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
(0 < Init.Nat.pred (Rlength (cons r1 (cons r2 r3))))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H21:r1 <= r2
pos_Rl l1 0 < (r1 + r2) / 2 < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
pos_Rl l1 0 < x < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  simpl; apply lt_O_Sn.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H21:r1 <= r2
pos_Rl l1 0 < (r1 + r2) / 2 < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
pos_Rl l1 0 < x < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  elim H21; intro.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H21:r1 <= r2
H22:r1 < r2
pos_Rl l1 0 < (r1 + r2) / 2 < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H21:r1 <= r2
H22:r1 = r2
pos_Rl l1 0 < (r1 + r2) / 2 < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
pos_Rl l1 0 < x < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  split.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H21:r1 <= r2
H22:r1 < r2
pos_Rl l1 0 < (r1 + r2) / 2
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H21:r1 <= r2
H22:r1 < r2
(r1 + r2) / 2 < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H21:r1 <= r2
H22:r1 = r2
pos_Rl l1 0 < (r1 + r2) / 2 < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
pos_Rl l1 0 < x < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite H17; simpl; apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
            | discrR ] ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H21:r1 <= r2
H22:r1 < r2
(r1 + r2) / 2 < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H21:r1 <= r2
H22:r1 = r2
pos_Rl l1 0 < (r1 + r2) / 2 < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
pos_Rl l1 0 < x < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite H17; simpl; apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite (Rplus_comm r1); rewrite double;
            apply Rplus_lt_compat_l; assumption
            | discrR ] ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H21:r1 <= r2
H22:r1 = r2
pos_Rl l1 0 < (r1 + r2) / 2 < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
pos_Rl l1 0 < x < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  elim H2; intros; rewrite H17 in H23; rewrite H17 in H24; simpl in H24;
    simpl in H23; rewrite H22 in H23;
      elim (Rlt_irrefl _ (Rlt_trans _ _ _ H23 H24)).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 0 < x < pos_Rl l1 1
H12:(2 <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0
H15:pos_Rl (cons_Rlist l1 l2) 1 = pos_Rl l1 1
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
H18:(0 < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
H19:(0 < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 0 < x0 < pos_Rl l1 1 ->
f x0 = pos_Rl lf1 0
pos_Rl l1 0 < x < pos_Rl l1 1
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl l1 i < x < pos_Rl l1 (S i) ->
(S (S i) <= Rlength l1)%nat ->
pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  clear Hreci; rewrite RList_P13.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
f x =
f
  ((pos_Rl (cons_Rlist (cons r2 r3) l2) i +
    pos_Rl (cons_Rlist (cons r2 r3) l2) (S i)) / 2)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite H17 in H14; rewrite H17 in H15;
    change
      (pos_Rl (cons_Rlist (cons r2 r3) l2) i =
        pos_Rl (cons r1 (cons r2 r3)) (S i)) in H14; rewrite H14;
      change
        (pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
          pos_Rl (cons r1 (cons r2 r3)) (S (S i))) in H15;
        rewrite H15; assert (H18 := H8 (S i));
          unfold constant_D_eq, open_interval in H18;
            assert (H19 : (S i < pred (Rlength l1))%nat).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
(S i < Init.Nat.pred (Rlength l1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
f x =
f
  ((pos_Rl (cons r1 (cons r2 r3)) (S i) +
    pos_Rl (cons r1 (cons r2 r3)) (S (S i))) / 2)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  apply lt_pred; apply lt_S_n; apply le_lt_n_Sm; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
f x =
f
  ((pos_Rl (cons r1 (cons r2 r3)) (S i) +
    pos_Rl (cons r1 (cons r2 r3)) (S (S i))) / 2)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  assert (H20 := H18 H19); repeat rewrite H20.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
pos_Rl lf1 (S i) = pos_Rl lf1 (S i)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
pos_Rl l1 (S i) <
(pos_Rl (cons r1 (cons r2 r3)) (S i) +
 pos_Rl (cons r1 (cons r2 r3)) (S (S i))) / 2 <
pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  reflexivity.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
pos_Rl l1 (S i) <
(pos_Rl (cons r1 (cons r2 r3)) (S i) +
 pos_Rl (cons r1 (cons r2 r3)) (S (S i))) / 2 <
pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite <- H17; assert (H21 : pos_Rl l1 (S i) <= pos_Rl l1 (S (S i))).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
pos_Rl l1 (S i) <= pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H21:pos_Rl l1 (S i) <= pos_Rl l1 (S (S i))
pos_Rl l1 (S i) <
(pos_Rl l1 (S i) + pos_Rl l1 (S (S i))) / 2 <
pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  apply (H3 (S i)); apply lt_pred; apply lt_S_n; apply le_lt_n_Sm; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H21:pos_Rl l1 (S i) <= pos_Rl l1 (S (S i))
pos_Rl l1 (S i) <
(pos_Rl l1 (S i) + pos_Rl l1 (S (S i))) / 2 <
pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  elim H21; intro.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H21:pos_Rl l1 (S i) <= pos_Rl l1 (S (S i))
H22:pos_Rl l1 (S i) < pos_Rl l1 (S (S i))
pos_Rl l1 (S i) <
(pos_Rl l1 (S i) + pos_Rl l1 (S (S i))) / 2 <
pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H21:pos_Rl l1 (S i) <= pos_Rl l1 (S (S i))
H22:pos_Rl l1 (S i) = pos_Rl l1 (S (S i))
pos_Rl l1 (S i) <
(pos_Rl l1 (S i) + pos_Rl l1 (S (S i))) / 2 <
pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  split.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H21:pos_Rl l1 (S i) <= pos_Rl l1 (S (S i))
H22:pos_Rl l1 (S i) < pos_Rl l1 (S (S i))
pos_Rl l1 (S i) <
(pos_Rl l1 (S i) + pos_Rl l1 (S (S i))) / 2
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H21:pos_Rl l1 (S i) <= pos_Rl l1 (S (S i))
H22:pos_Rl l1 (S i) < pos_Rl l1 (S (S i))
(pos_Rl l1 (S i) + pos_Rl l1 (S (S i))) / 2 <
pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H21:pos_Rl l1 (S i) <= pos_Rl l1 (S (S i))
H22:pos_Rl l1 (S i) = pos_Rl l1 (S (S i))
pos_Rl l1 (S i) <
(pos_Rl l1 (S i) + pos_Rl l1 (S (S i))) / 2 <
pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
            | discrR ] ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H21:pos_Rl l1 (S i) <= pos_Rl l1 (S (S i))
H22:pos_Rl l1 (S i) < pos_Rl l1 (S (S i))
(pos_Rl l1 (S i) + pos_Rl l1 (S (S i))) / 2 <
pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H21:pos_Rl l1 (S i) <= pos_Rl l1 (S (S i))
H22:pos_Rl l1 (S i) = pos_Rl l1 (S (S i))
pos_Rl l1 (S i) <
(pos_Rl l1 (S i) + pos_Rl l1 (S (S i))) / 2 <
pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite (Rplus_comm (pos_Rl l1 (S i)));
            rewrite double; apply Rplus_lt_compat_l; assumption
            | discrR ] ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H21:pos_Rl l1 (S i) <= pos_Rl l1 (S (S i))
H22:pos_Rl l1 (S i) = pos_Rl l1 (S (S i))
pos_Rl l1 (S i) <
(pos_Rl l1 (S i) + pos_Rl l1 (S (S i))) / 2 <
pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  elim H2; intros; rewrite H22 in H23;
    elim (Rlt_irrefl _ (Rlt_trans _ _ _ H23 H24)).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H14:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl (cons r1 (cons r2 r3)) (S i)
H15:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl (cons r1 (cons r2 r3)) (S (S i))
H16:(2 <= Rlength l1)%nat
H17:l1 = cons r1 (cons r2 r3)
H18:(S i < Init.Nat.pred (Rlength l1))%nat ->
forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
H19:(S i < Init.Nat.pred (Rlength l1))%nat
H20:forall x0 : R,
pos_Rl l1 (S i) < x0 < pos_Rl l1 (S (S i)) ->
f x0 = pos_Rl lf1 (S i)
pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 (S i) < x < pos_Rl l1 (S (S i))
H12:(S (S (S i)) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H15:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l1 (S (S i))
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  simpl; rewrite H17 in H1; simpl in H1; apply lt_S_n; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl l1 i < x < pos_Rl l1 (S i)
H12:(S (S i) <= Rlength l1)%nat
H14:pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i
H15:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l1 (S i)
H16:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H17:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite RList_P14; rewrite H17 in H1; simpl in H1; apply H1.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  inversion H12.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  assert (H16 : pos_Rl (cons_Rlist l1 l2) (S i) = b).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
pos_Rl (cons_Rlist l1 l2) (S i) = b
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite RList_P29.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
pos_Rl l2 (S i - Rlength l1) = b
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
(Rlength l1 <= S i)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
(S i < Rlength (cons_Rlist l1 l2))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite H15; rewrite <- minus_n_n; rewrite H10; unfold Rmin;
    case (Rle_dec b c) as [|[]]; [ reflexivity | left; assumption ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
(Rlength l1 <= S i)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
(S i < Rlength (cons_Rlist l1 l2))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite H15; apply le_n.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
(S i < Rlength (cons_Rlist l1 l2))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  induction  l1 as [| r l1 Hrecl1].f:R -> R
a, b, c:R
l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist nil
H5:pos_Rl nil 0 = Rmin a b
H4:pos_Rl nil (Init.Nat.pred (Rlength nil)) =
Rmax a b
H6:Rlength nil = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq f
  (open_interval (pos_Rl nil i0)
     (pos_Rl nil (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist nil l2)))%nat
x:R
H2:pos_Rl (cons_Rlist nil l2) i < x <
pos_Rl (cons_Rlist nil l2) (S i)
H12:(Rlength nil < S (S i))%nat
H15:Rlength nil = S i
(S i < Rlength (cons_Rlist nil l2))%nat
f:R -> R
a, b, c, r:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist (cons r l1)
H5:pos_Rl (cons r l1) 0 = Rmin a b
H4:pos_Rl (cons r l1)
  (Init.Nat.pred (Rlength (cons r l1))) =
Rmax a b
H6:Rlength (cons r l1) = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l1)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l1) i0)
     (pos_Rl (cons r l1) (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i <
 Init.Nat.pred
   (Rlength (cons_Rlist (cons r l1) l2)))%nat
x:R
H2:pos_Rl (cons_Rlist (cons r l1) l2) i < x <
pos_Rl (cons_Rlist (cons r l1) l2) (S i)
H12:(Rlength (cons r l1) < S (S i))%nat
H15:Rlength (cons r l1) = S i
Hrecl1:ordered_Rlist l1 ->
pos_Rl l1 0 = Rmin a b ->
pos_Rl l1 (Init.Nat.pred (Rlength l1)) =
Rmax a b ->
Rlength l1 = S (Rlength lf1) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l1))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l1 i0)
      (pos_Rl l1 (S i0))) 
   (pos_Rl lf1 i0)) ->
(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i) ->
(Rlength l1 < S (S i))%nat ->
Rlength l1 = S i ->
(S i < Rlength (cons_Rlist l1 l2))%nat
(S i < Rlength (cons_Rlist (cons r l1) l2))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  simpl in H15; discriminate.f:R -> R
a, b, c, r:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist (cons r l1)
H5:pos_Rl (cons r l1) 0 = Rmin a b
H4:pos_Rl (cons r l1)
  (Init.Nat.pred (Rlength (cons r l1))) =
Rmax a b
H6:Rlength (cons r l1) = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l1)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l1) i0)
     (pos_Rl (cons r l1) (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i <
 Init.Nat.pred
   (Rlength (cons_Rlist (cons r l1) l2)))%nat
x:R
H2:pos_Rl (cons_Rlist (cons r l1) l2) i < x <
pos_Rl (cons_Rlist (cons r l1) l2) (S i)
H12:(Rlength (cons r l1) < S (S i))%nat
H15:Rlength (cons r l1) = S i
Hrecl1:ordered_Rlist l1 ->
pos_Rl l1 0 = Rmin a b ->
pos_Rl l1 (Init.Nat.pred (Rlength l1)) =
Rmax a b ->
Rlength l1 = S (Rlength lf1) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l1))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l1 i0)
      (pos_Rl l1 (S i0))) 
   (pos_Rl lf1 i0)) ->
(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i) ->
(Rlength l1 < S (S i))%nat ->
Rlength l1 = S i ->
(S i < Rlength (cons_Rlist l1 l2))%nat
(S i < Rlength (cons_Rlist (cons r l1) l2))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  clear Hrecl1; simpl in H1; simpl; apply lt_n_S; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  assert (H17 : pos_Rl (cons_Rlist l1 l2) i = b).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
pos_Rl (cons_Rlist l1 l2) i = b
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
H17:pos_Rl (cons_Rlist l1 l2) i = b
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite RList_P26.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
pos_Rl l1 i = b
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
(i < Rlength l1)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
H17:pos_Rl (cons_Rlist l1 l2) i = b
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  replace i with (pred (Rlength l1));
  [ rewrite H4; unfold Rmax; case (Rle_dec a b) as [|[]];
    [ reflexivity | left; assumption ]
    | rewrite H15; reflexivity ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
(i < Rlength l1)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
H17:pos_Rl (cons_Rlist l1 l2) i = b
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite H15; apply lt_n_Sn.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
H15:Rlength l1 = S i
H16:pos_Rl (cons_Rlist l1 l2) (S i) = b
H17:pos_Rl (cons_Rlist l1 l2) i = b
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite H16 in H2; rewrite H17 in H2; elim H2; intros;
    elim (Rlt_irrefl _ (Rlt_trans _ _ _ H14 H18)).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  assert (H16 : pos_Rl (cons_Rlist l1 l2) i = pos_Rl l2 (i - Rlength l1)).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  apply RList_P29.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
(Rlength l1 <= i)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
(i < Rlength (cons_Rlist l1 l2))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  apply le_S_n; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
(i < Rlength (cons_Rlist l1 l2))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  apply lt_le_trans with (pred (Rlength (cons_Rlist l1 l2)));
    [ assumption | apply le_pred_n ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  assert
    (H17 : pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l2 (S (i - Rlength l1))).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  replace (S (i - Rlength l1)) with (S i - Rlength l1)%nat.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
(S i - Rlength l1)%nat = S (i - Rlength l1)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  apply RList_P29.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
(Rlength l1 <= S i)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
(S i < Rlength (cons_Rlist l1 l2))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
(S i - Rlength l1)%nat = S (i - Rlength l1)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  apply le_S_n; apply le_trans with (S i); [ assumption | apply le_n_Sn ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
(S i < Rlength (cons_Rlist l1 l2))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
(S i - Rlength l1)%nat = S (i - Rlength l1)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  induction  l1 as [| r l1 Hrecl1].f:R -> R
a, b, c:R
l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist nil
H5:pos_Rl nil 0 = Rmin a b
H4:pos_Rl nil (Init.Nat.pred (Rlength nil)) =
Rmax a b
H6:Rlength nil = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq f
  (open_interval (pos_Rl nil i0)
     (pos_Rl nil (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist nil l2)))%nat
x:R
H2:pos_Rl (cons_Rlist nil l2) i < x <
pos_Rl (cons_Rlist nil l2) (S i)
H12:(Rlength nil < S (S i))%nat
m:nat
H15:(S (Rlength nil) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist nil l2) i =
pos_Rl l2 (i - Rlength nil)
(S i < Rlength (cons_Rlist nil l2))%nat
f:R -> R
a, b, c, r:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist (cons r l1)
H5:pos_Rl (cons r l1) 0 = Rmin a b
H4:pos_Rl (cons r l1)
  (Init.Nat.pred (Rlength (cons r l1))) =
Rmax a b
H6:Rlength (cons r l1) = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l1)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l1) i0)
     (pos_Rl (cons r l1) (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i <
 Init.Nat.pred
   (Rlength (cons_Rlist (cons r l1) l2)))%nat
x:R
H2:pos_Rl (cons_Rlist (cons r l1) l2) i < x <
pos_Rl (cons_Rlist (cons r l1) l2) (S i)
H12:(Rlength (cons r l1) < S (S i))%nat
m:nat
H15:(S (Rlength (cons r l1)) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist (cons r l1) l2) i =
pos_Rl l2 (i - Rlength (cons r l1))
Hrecl1:ordered_Rlist l1 ->
pos_Rl l1 0 = Rmin a b ->
pos_Rl l1 (Init.Nat.pred (Rlength l1)) =
Rmax a b ->
Rlength l1 = S (Rlength lf1) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l1))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l1 i0)
      (pos_Rl l1 (S i0))) 
   (pos_Rl lf1 i0)) ->
(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i) ->
(Rlength l1 < S (S i))%nat ->
(S (Rlength l1) <= S i)%nat ->
pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1) ->
(S i < Rlength (cons_Rlist l1 l2))%nat
(S i < Rlength (cons_Rlist (cons r l1) l2))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
(S i - Rlength l1)%nat = S (i - Rlength l1)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  simpl in H6; discriminate.f:R -> R
a, b, c, r:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist (cons r l1)
H5:pos_Rl (cons r l1) 0 = Rmin a b
H4:pos_Rl (cons r l1)
  (Init.Nat.pred (Rlength (cons r l1))) =
Rmax a b
H6:Rlength (cons r l1) = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r l1)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r l1) i0)
     (pos_Rl (cons r l1) (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i <
 Init.Nat.pred
   (Rlength (cons_Rlist (cons r l1) l2)))%nat
x:R
H2:pos_Rl (cons_Rlist (cons r l1) l2) i < x <
pos_Rl (cons_Rlist (cons r l1) l2) (S i)
H12:(Rlength (cons r l1) < S (S i))%nat
m:nat
H15:(S (Rlength (cons r l1)) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist (cons r l1) l2) i =
pos_Rl l2 (i - Rlength (cons r l1))
Hrecl1:ordered_Rlist l1 ->
pos_Rl l1 0 = Rmin a b ->
pos_Rl l1 (Init.Nat.pred (Rlength l1)) =
Rmax a b ->
Rlength l1 = S (Rlength lf1) ->
(forall i0 : nat,
 (i0 < Init.Nat.pred (Rlength l1))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l1 i0)
      (pos_Rl l1 (S i0))) 
   (pos_Rl lf1 i0)) ->
(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i) ->
(Rlength l1 < S (S i))%nat ->
(S (Rlength l1) <= S i)%nat ->
pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1) ->
(S i < Rlength (cons_Rlist l1 l2))%nat
(S i < Rlength (cons_Rlist (cons r l1) l2))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
(S i - Rlength l1)%nat = S (i - Rlength l1)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  clear Hrecl1; simpl in H1; simpl; apply lt_n_S; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
(S i - Rlength l1)%nat = S (i - Rlength l1)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  symmetry ; apply minus_Sn_m; apply le_S_n; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  assert (H18 : (2 <= Rlength l1)%nat).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
(2 <= Rlength l1)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  clear f c l2 lf2 H0 H3 H8 H7 H10 H9 H11 H13 i H1 x H2 H12 m H14 H15 H16 H17;
    induction  l1 as [| r l1 Hrecl1].a, b:R
lf1:Rlist
H:a < b
H5:pos_Rl nil 0 = Rmin a b
H4:pos_Rl nil (Init.Nat.pred (Rlength nil)) =
Rmax a b
H6:Rlength nil = S (Rlength lf1)
(2 <= Rlength nil)%nat
a, b, r:R
l1, lf1:Rlist
H:a < b
H5:pos_Rl (cons r l1) 0 = Rmin a b
H4:pos_Rl (cons r l1)
  (Init.Nat.pred (Rlength (cons r l1))) =
Rmax a b
H6:Rlength (cons r l1) = S (Rlength lf1)
Hrecl1:pos_Rl l1 0 = Rmin a b ->
pos_Rl l1 (Init.Nat.pred (Rlength l1)) =
Rmax a b ->
Rlength l1 = S (Rlength lf1) ->
(2 <= Rlength l1)%nat
(2 <= Rlength (cons r l1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  discriminate.a, b, r:R
l1, lf1:Rlist
H:a < b
H5:pos_Rl (cons r l1) 0 = Rmin a b
H4:pos_Rl (cons r l1)
  (Init.Nat.pred (Rlength (cons r l1))) =
Rmax a b
H6:Rlength (cons r l1) = S (Rlength lf1)
Hrecl1:pos_Rl l1 0 = Rmin a b ->
pos_Rl l1 (Init.Nat.pred (Rlength l1)) =
Rmax a b ->
Rlength l1 = S (Rlength lf1) ->
(2 <= Rlength l1)%nat
(2 <= Rlength (cons r l1))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  clear Hrecl1; induction  l1 as [| r0 l1 Hrecl1].a, b, r:R
lf1:Rlist
H:a < b
H5:pos_Rl (cons r nil) 0 = Rmin a b
H4:pos_Rl (cons r nil)
  (Init.Nat.pred (Rlength (cons r nil))) =
Rmax a b
H6:Rlength (cons r nil) = S (Rlength lf1)
(2 <= Rlength (cons r nil))%nat
a, b, r, r0:R
l1, lf1:Rlist
H:a < b
H5:pos_Rl (cons r (cons r0 l1)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r0 l1))
  (Init.Nat.pred (Rlength (cons r (cons r0 l1)))) =
Rmax a b
H6:Rlength (cons r (cons r0 l1)) = S (Rlength lf1)
Hrecl1:pos_Rl (cons r l1) 0 = Rmin a b ->
pos_Rl (cons r l1)
  (Init.Nat.pred (Rlength (cons r l1))) =
Rmax a b ->
Rlength (cons r l1) = S (Rlength lf1) ->
(2 <= Rlength (cons r l1))%nat
(2 <= Rlength (cons r (cons r0 l1)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  simpl in H5; simpl in H4; assert (H0 : Rmin a b < Rmax a b).a, b, r:R
lf1:Rlist
H:a < b
H5:r = Rmin a b
H4:r = Rmax a b
H6:Rlength (cons r nil) = S (Rlength lf1)
Rmin a b < Rmax a b
a, b, r:R
lf1:Rlist
H:a < b
H5:r = Rmin a b
H4:r = Rmax a b
H6:Rlength (cons r nil) = S (Rlength lf1)
H0:Rmin a b < Rmax a b
(2 <= Rlength (cons r nil))%nat
a, b, r, r0:R
l1, lf1:Rlist
H:a < b
H5:pos_Rl (cons r (cons r0 l1)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r0 l1))
  (Init.Nat.pred (Rlength (cons r (cons r0 l1)))) =
Rmax a b
H6:Rlength (cons r (cons r0 l1)) = S (Rlength lf1)
Hrecl1:pos_Rl (cons r l1) 0 = Rmin a b ->
pos_Rl (cons r l1)
  (Init.Nat.pred (Rlength (cons r l1))) =
Rmax a b ->
Rlength (cons r l1) = S (Rlength lf1) ->
(2 <= Rlength (cons r l1))%nat
(2 <= Rlength (cons r (cons r0 l1)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  unfold Rmin, Rmax; case (Rle_dec a b) as [|[]];
    [ assumption | left; assumption ].a, b, r:R
lf1:Rlist
H:a < b
H5:r = Rmin a b
H4:r = Rmax a b
H6:Rlength (cons r nil) = S (Rlength lf1)
H0:Rmin a b < Rmax a b
(2 <= Rlength (cons r nil))%nat
a, b, r, r0:R
l1, lf1:Rlist
H:a < b
H5:pos_Rl (cons r (cons r0 l1)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r0 l1))
  (Init.Nat.pred (Rlength (cons r (cons r0 l1)))) =
Rmax a b
H6:Rlength (cons r (cons r0 l1)) = S (Rlength lf1)
Hrecl1:pos_Rl (cons r l1) 0 = Rmin a b ->
pos_Rl (cons r l1)
  (Init.Nat.pred (Rlength (cons r l1))) =
Rmax a b ->
Rlength (cons r l1) = S (Rlength lf1) ->
(2 <= Rlength (cons r l1))%nat
(2 <= Rlength (cons r (cons r0 l1)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  rewrite <- H5 in H0; rewrite <- H4 in H0; elim (Rlt_irrefl _ H0).a, b, r, r0:R
l1, lf1:Rlist
H:a < b
H5:pos_Rl (cons r (cons r0 l1)) 0 = Rmin a b
H4:pos_Rl (cons r (cons r0 l1))
  (Init.Nat.pred (Rlength (cons r (cons r0 l1)))) =
Rmax a b
H6:Rlength (cons r (cons r0 l1)) = S (Rlength lf1)
Hrecl1:pos_Rl (cons r l1) 0 = Rmin a b ->
pos_Rl (cons r l1)
  (Init.Nat.pred (Rlength (cons r l1))) =
Rmax a b ->
Rlength (cons r l1) = S (Rlength lf1) ->
(2 <= Rlength (cons r l1))%nat
(2 <= Rlength (cons r (cons r0 l1)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  clear Hrecl1; simpl; repeat apply le_n_S; apply le_O_n.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
f x = pos_Rl (FF (cons_Rlist l1 l2) f) i
  elim (RList_P20 _ H18); intros r1 [r2 [r3 H19]]; rewrite H19;
    change
      (f x = pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i)
     ; rewrite RList_P12.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     i)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  induction  i as [| i Hreci].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i : nat,
(i < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i) (pos_Rl l1 (S i)))
  (pos_Rl lf1 i)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H1:(0 < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) 0 < x <
pos_Rl (cons_Rlist l1 l2) 1
H12:(Rlength l1 < 2)%nat
m:nat
H15:(S (Rlength l1) <= 1)%nat
H14:m = 1%nat
H16:pos_Rl (cons_Rlist l1 l2) 0 =
pos_Rl l2 (0 - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) 1 =
pos_Rl l2 (S (0 - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     0)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i) ->
(Rlength l1 < S (S i))%nat ->
(S (Rlength l1) <= S i)%nat ->
m = S i ->
pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1) ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1)) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  assert (H20 := le_S_n _ _ H15); assert (H21 := le_trans _ _ _ H18 H20);
    elim (le_Sn_O _ H21).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
Hreci:(i <
 Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat ->
pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i) ->
(Rlength l1 < S (S i))%nat ->
(S (Rlength l1) <= S i)%nat ->
m = S i ->
pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1) ->
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1)) ->
f x =
f
  (pos_Rl
     (mid_Rlist (cons_Rlist (cons r2 r3) l2)
        r1) i)
f x =
f
  (pos_Rl (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1)
     (S i))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  clear Hreci; rewrite RList_P13.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
f x =
f
  ((pos_Rl (cons_Rlist (cons r2 r3) l2) i +
    pos_Rl (cons_Rlist (cons r2 r3) l2) (S i)) / 2)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  rewrite H19 in H16; rewrite H19 in H17;
    change
      (pos_Rl (cons_Rlist (cons r2 r3) l2) i =
        pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3))))
      in H16; rewrite H16;
        change
          (pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
            pos_Rl l2 (S (S i - Rlength (cons r1 (cons r2 r3)))))
          in H17; rewrite H17; assert (H20 := H13 (S i - Rlength l1)%nat);
            unfold constant_D_eq, open_interval in H20;
              assert (H21 : (S i - Rlength l1 < pred (Rlength l2))%nat).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
f x =
f
  ((pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3))) +
    pos_Rl l2
      (S (S i - Rlength (cons r1 (cons r2 r3))))) / 2)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  apply lt_pred; rewrite minus_Sn_m.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(S (S i) - Rlength l1 < Rlength l2)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(Rlength l1 <= S i)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
f x =
f
  ((pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3))) +
    pos_Rl l2
      (S (S i - Rlength (cons r1 (cons r2 r3))))) / 2)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  apply plus_lt_reg_l with (Rlength l1); rewrite <- le_plus_minus.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(S (S i) < Rlength l1 + Rlength l2)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(Rlength l1 <= S (S i))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(Rlength l1 <= S i)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
f x =
f
  ((pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3))) +
    pos_Rl l2
      (S (S i - Rlength (cons r1 (cons r2 r3))))) / 2)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  rewrite H19 in H1; simpl in H1; rewrite H19; simpl;
    rewrite RList_P23 in H1; apply lt_n_S; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(Rlength l1 <= S (S i))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(Rlength l1 <= S i)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
f x =
f
  ((pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3))) +
    pos_Rl l2
      (S (S i - Rlength (cons r1 (cons r2 r3))))) / 2)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  apply le_trans with (S i); [ apply le_S_n; assumption | apply le_n_Sn ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(Rlength l1 <= S i)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
f x =
f
  ((pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3))) +
    pos_Rl l2
      (S (S i - Rlength (cons r1 (cons r2 r3))))) / 2)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  apply le_S_n; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
f x =
f
  ((pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3))) +
    pos_Rl l2
      (S (S i - Rlength (cons r1 (cons r2 r3))))) / 2)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  assert (H22 := H20 H21); repeat rewrite H22.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl lf2 (S i - Rlength l1) =
pos_Rl lf2 (S i - Rlength l1)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3))) +
 pos_Rl l2 (S (S i - Rlength (cons r1 (cons r2 r3))))) /
2 < pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  reflexivity.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3))) +
 pos_Rl l2 (S (S i - Rlength (cons r1 (cons r2 r3))))) /
2 < pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  rewrite <- H19;
    assert
      (H23 : pos_Rl l2 (S i - Rlength l1) <= pos_Rl l2 (S (S i - Rlength l1))).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  apply H7; apply lt_pred.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(S (S i - Rlength l1) < Rlength l2)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  rewrite minus_Sn_m.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(S (S i) - Rlength l1 < Rlength l2)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(Rlength l1 <= S i)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  apply plus_lt_reg_l with (Rlength l1); rewrite <- le_plus_minus.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(S (S i) < Rlength l1 + Rlength l2)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(Rlength l1 <= S (S i))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(Rlength l1 <= S i)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  rewrite H19 in H1; simpl in H1; rewrite H19; simpl;
    rewrite RList_P23 in H1; apply lt_n_S; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(Rlength l1 <= S (S i))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(Rlength l1 <= S i)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  apply le_trans with (S i); [ apply le_S_n; assumption | apply le_n_Sn ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
(Rlength l1 <= S i)%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  apply le_S_n; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  elim H23; intro.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
H24:pos_Rl l2 (S i - Rlength l1) <
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
H24:pos_Rl l2 (S i - Rlength l1) =
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  split.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
H24:pos_Rl l2 (S i - Rlength l1) <
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
H24:pos_Rl l2 (S i - Rlength l1) <
pos_Rl l2 (S (S i - Rlength l1))
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
H24:pos_Rl l2 (S i - Rlength l1) =
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
            | discrR ] ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
H24:pos_Rl l2 (S i - Rlength l1) <
pos_Rl l2 (S (S i - Rlength l1))
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
H24:pos_Rl l2 (S i - Rlength l1) =
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  apply Rmult_lt_reg_l with 2;
    [ prove_sup0
      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
        rewrite <- Rinv_r_sym;
          [ rewrite Rmult_1_l; rewrite (Rplus_comm (pos_Rl l2 (S i - Rlength l1)));
            rewrite double; apply Rplus_lt_compat_l; assumption
            | discrR ] ].f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl l2 (S i - Rlength l1) <=
pos_Rl l2 (S (S i - Rlength l1))
H24:pos_Rl l2 (S i - Rlength l1) =
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) <
(pos_Rl l2 (S i - Rlength l1) +
 pos_Rl l2 (S (S i - Rlength l1))) / 2 <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  rewrite <- H19 in H16; rewrite <- H19 in H17; elim H2; intros;
    rewrite H19 in H25; rewrite H19 in H26; simpl in H25;
      simpl in H16; rewrite H16 in H25; simpl in H26; simpl in H17;
        rewrite H17 in H26; simpl in H24; rewrite H24 in H25;
          elim (Rlt_irrefl _ (Rlt_trans _ _ _ H25 H26)).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  assert (H23 : pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l2 (S i - Rlength l1)).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  rewrite H19; simpl; simpl in H16; apply H16.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  assert
    (H24 :
      pos_Rl (cons_Rlist l1 l2) (S (S i)) = pos_Rl l2 (S (S i - Rlength l1))).f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H24:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  rewrite H19; simpl; simpl in H17; apply H17.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
r1, r2:R
r3:Rlist
H16:pos_Rl (cons_Rlist (cons r2 r3) l2) i =
pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))
H17:pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) =
pos_Rl l2
  (S (S i - Rlength (cons r1 (cons r2 r3))))
H18:(2 <= Rlength l1)%nat
H19:l1 = cons r1 (cons r2 r3)
H20:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat ->
forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H21:(S i - Rlength l1 < Init.Nat.pred (Rlength l2))%nat
H22:forall x0 : R,
pos_Rl l2 (S i - Rlength l1) < x0 <
pos_Rl l2 (S (S i - Rlength l1)) ->
f x0 = pos_Rl lf2 (S i - Rlength l1)
H23:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H24:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
pos_Rl l2 (S i - Rlength l1) < x <
pos_Rl l2 (S (S i - Rlength l1))
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  rewrite <- H23; rewrite <- H24; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(S i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) (S i) < x <
pos_Rl (cons_Rlist l1 l2) (S (S i))
H12:(Rlength l1 < S (S (S i)))%nat
m:nat
H15:(S (Rlength l1) <= S (S i))%nat
H14:m = S (S i)
H16:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S (S i)) =
pos_Rl l2 (S (S i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Init.Nat.pred (Rlength (cons_Rlist (cons r2 r3) l2)))%nat
f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  simpl; rewrite H19 in H1; simpl in H1; apply lt_S_n; assumption.f:R -> R
a, b, c:R
l1, l2, lf1, lf2:Rlist
H:a < b
H0:b < c
H3:ordered_Rlist l1
H5:pos_Rl l1 0 = Rmin a b
H4:pos_Rl l1 (Init.Nat.pred (Rlength l1)) = Rmax a b
H6:Rlength l1 = S (Rlength lf1)
H8:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l1))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1 i0)
     (pos_Rl l1 (S i0))) 
  (pos_Rl lf1 i0)
H7:ordered_Rlist l2
H10:pos_Rl l2 0 = Rmin b c
H9:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H11:Rlength l2 = S (Rlength lf2)
H13:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i0)
     (pos_Rl l2 (S i0))) 
  (pos_Rl lf2 i0)
i:nat
H1:(i < Init.Nat.pred (Rlength (cons_Rlist l1 l2)))%nat
x:R
H2:pos_Rl (cons_Rlist l1 l2) i < x <
pos_Rl (cons_Rlist l1 l2) (S i)
H12:(Rlength l1 < S (S i))%nat
m:nat
H15:(S (Rlength l1) <= S i)%nat
H14:m = S i
H16:pos_Rl (cons_Rlist l1 l2) i =
pos_Rl l2 (i - Rlength l1)
H17:pos_Rl (cons_Rlist l1 l2) (S i) =
pos_Rl l2 (S (i - Rlength l1))
H18:(2 <= Rlength l1)%nat
r1, r2:R
r3:Rlist
H19:l1 = cons r1 (cons r2 r3)
(i <
 Rlength (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1))%nat
  rewrite RList_P14; rewrite H19 in H1; simpl in H1; simpl; apply H1.
Qed.

Lemma StepFun_P41 :
  forall (f:R -> R) (a b c:R),
    a <= b -> b <= c -> IsStepFun f a b -> IsStepFun f b c -> IsStepFun f a c.forall (f : R -> R) (a b c : R),
a <= b ->
b <= c ->
IsStepFun f a b -> IsStepFun f b c -> IsStepFun f a c
Proof.forall (f : R -> R) (a b c : R),
a <= b ->
b <= c ->
IsStepFun f a b -> IsStepFun f b c -> IsStepFun f a c
  intros f a b c H H0 (l1,(lf1,H1)) (l2,(lf2,H2));
    destruct (total_order_T a b) as [[Hltab|Hab]|Hgtab].f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hltab:a < b
IsStepFun f a c
f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hab:a = b
IsStepFun f a c
f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hgtab:a > b
IsStepFun f a c
  destruct (total_order_T b c) as [[Hltbc|Hbc]|Hgtbc].f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hltab:a < b
Hltbc:b < c
IsStepFun f a c
f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hltab:a < b
Hbc:b = c
IsStepFun f a c
f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hltab:a < b
Hgtbc:b > c
IsStepFun f a c
f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hab:a = b
IsStepFun f a c
f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hgtab:a > b
IsStepFun f a c
  exists (cons_Rlist l1 l2); exists (FF (cons_Rlist l1 l2) f);
    apply StepFun_P40 with b lf1 lf2; assumption.f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hltab:a < b
Hbc:b = c
IsStepFun f a c
f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hltab:a < b
Hgtbc:b > c
IsStepFun f a c
f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hab:a = b
IsStepFun f a c
f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hgtab:a > b
IsStepFun f a c
  exists l1; exists lf1; rewrite Hbc in H1; assumption.f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hltab:a < b
Hgtbc:b > c
IsStepFun f a c
f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hab:a = b
IsStepFun f a c
f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hgtab:a > b
IsStepFun f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 Hgtbc)).f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hab:a = b
IsStepFun f a c
f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hgtab:a > b
IsStepFun f a c
  exists l2; exists lf2; rewrite <- Hab in H2; assumption.f:R -> R
a, b, c:R
H:a <= b
H0:b <= c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
Hgtab:a > b
IsStepFun f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H Hgtab)).
Qed.

Lemma StepFun_P42 :
  forall (l1 l2:Rlist) (f:R -> R),
    pos_Rl l1 (pred (Rlength l1)) = pos_Rl l2 0 ->
    Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2) =
    Int_SF (FF l1 f) l1 + Int_SF (FF l2 f) l2.forall (l1 l2 : Rlist) (f : R -> R),
pos_Rl l1 (Init.Nat.pred (Rlength l1)) = pos_Rl l2 0 ->
Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2) =
Int_SF (FF l1 f) l1 + Int_SF (FF l2 f) l2
Proof.forall (l1 l2 : Rlist) (f : R -> R),
pos_Rl l1 (Init.Nat.pred (Rlength l1)) = pos_Rl l2 0 ->
Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2) =
Int_SF (FF l1 f) l1 + Int_SF (FF l2 f) l2
  intros l1 l2 f; induction l1 as [| r l1 IHl1]; intros H;
    [ simpl; ring
      | destruct l1 as [| r0 r1];
        [ simpl in H; simpl; destruct l2 as [| r0 r1];
          [ simpl; ring | simpl; simpl in H; rewrite H; ring ]
          | simpl; rewrite Rplus_assoc; apply Rplus_eq_compat_l; apply IHl1;
            rewrite <- H; reflexivity ] ].
Qed.

Lemma StepFun_P43 :
  forall (f:R -> R) (a b c:R) (pr1:IsStepFun f a b)
    (pr2:IsStepFun f b c) (pr3:IsStepFun f a c),
    RiemannInt_SF (mkStepFun pr1) + RiemannInt_SF (mkStepFun pr2) =
    RiemannInt_SF (mkStepFun pr3).forall (f : R -> R) (a b c : R)
  (pr1 : IsStepFun f a b) (pr2 : IsStepFun f b c)
  (pr3 : IsStepFun f a c),
RiemannInt_SF {| fe := f; pre := pr1 |} +
RiemannInt_SF {| fe := f; pre := pr2 |} =
RiemannInt_SF {| fe := f; pre := pr3 |}
Proof.forall (f : R -> R) (a b c : R)
  (pr1 : IsStepFun f a b) (pr2 : IsStepFun f b c)
  (pr3 : IsStepFun f a c),
RiemannInt_SF {| fe := f; pre := pr1 |} +
RiemannInt_SF {| fe := f; pre := pr2 |} =
RiemannInt_SF {| fe := f; pre := pr3 |}
  intros f; intros.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
RiemannInt_SF {| fe := f; pre := pr1 |} +
RiemannInt_SF {| fe := f; pre := pr2 |} =
RiemannInt_SF {| fe := f; pre := pr3 |}
  pose proof pr1 as (l1,(lf1,H1)).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
RiemannInt_SF {| fe := f; pre := pr1 |} +
RiemannInt_SF {| fe := f; pre := pr2 |} =
RiemannInt_SF {| fe := f; pre := pr3 |}
  pose proof pr2 as (l2,(lf2,H2)).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
RiemannInt_SF {| fe := f; pre := pr1 |} +
RiemannInt_SF {| fe := f; pre := pr2 |} =
RiemannInt_SF {| fe := f; pre := pr3 |}
  pose proof pr3 as (l3,(lf3,H3)).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
RiemannInt_SF {| fe := f; pre := pr1 |} +
RiemannInt_SF {| fe := f; pre := pr2 |} =
RiemannInt_SF {| fe := f; pre := pr3 |}
  replace (RiemannInt_SF (mkStepFun pr1)) with
  match Rle_dec a b with
    | left _ => Int_SF lf1 l1
    | right _ => - Int_SF lf1 l1
  end.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) +
RiemannInt_SF {| fe := f; pre := pr2 |} =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (RiemannInt_SF (mkStepFun pr2)) with
  match Rle_dec b c with
    | left _ => Int_SF lf2 l2
    | right _ => - Int_SF lf2 l2
  end.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) +
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (RiemannInt_SF (mkStepFun pr3)) with
  match Rle_dec a c with
    | left _ => Int_SF lf3 l3
    | right _ => - Int_SF lf3 l3
  end.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) +
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  case (Rle_dec a b) as [Hle|Hnle]; case (Rle_dec b c) as [Hle'|Hnle']; case (Rle_dec a c) as [Hle''|Hnle''].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  elim Hle; intro.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  elim Hle'; intro.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b = c
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf3 l3) with
  (Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2)).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF lf1 l1 + Int_SF lf2 l2 =
Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b = c
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF l1 f) l1 + Int_SF lf2 l2 =
Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b = c
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF l1 f) l1 + Int_SF (FF l2 f) l2 =
Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF l2 f) l2 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b = c
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  symmetry ; apply StepFun_P42.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
pos_Rl l1 (Init.Nat.pred (Rlength l1)) = pos_Rl l2 0
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF l2 f) l2 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b = c
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  unfold adapted_couple in H1, H2; decompose [and] H1; decompose [and] H2;
    clear H1 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin;
      decide (Rle_dec a b) with Hle; decide (Rle_dec b c) with Hle'; reflexivity.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF l2 f) l2 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b = c
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17;
    [ apply StepFun_P21; unfold is_subdivision; split with lf2; apply H2;
      assumption
      | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b = c
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17;
    [ apply StepFun_P21; unfold is_subdivision; split with lf1; apply H1
      | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b < c
Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b = c
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17; [ apply (StepFun_P40 H H0 H1 H2) | apply H3 ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b = c
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf2 l2) with 0.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b = c
Int_SF lf1 l1 + 0 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b = c
0 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  rewrite Rplus_0_r; eapply StepFun_P17;
    [ apply H1 | rewrite <- H0 in H3; apply H3 ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a < b
H0:b = c
0 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  symmetry ; eapply StepFun_P8; [ apply H2 | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf1 l1) with 0.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
0 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
0 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  rewrite Rplus_0_l; eapply StepFun_P17;
    [ apply H2 | rewrite H in H3; apply H3 ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hle'':a <= c
H:a = b
0 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  symmetry ; eapply StepFun_P8; [ apply H1 | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hle':b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  elim Hnle''; apply Rle_trans with b; assumption.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply Rplus_eq_reg_l with (Int_SF lf2 l2);
    replace (Int_SF lf2 l2 + (Int_SF lf1 l1 + - Int_SF lf2 l2)) with
    (Int_SF lf1 l1); [ idtac | ring ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  assert (H : c < b).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
c < b
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  auto with real.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  elim Hle''; intro.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a = c
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  rewrite Rplus_comm;
    replace (Int_SF lf1 l1) with
    (Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2)).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2) =
Int_SF lf3 l3 + Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a = c
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2) =
Int_SF (FF l3 f) l3 + Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a = c
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2) =
Int_SF (FF l3 f) l3 + Int_SF (FF l2 f) l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF l2 f) l2 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a = c
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply StepFun_P42.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
pos_Rl l3 (Init.Nat.pred (Rlength l3)) = pos_Rl l2 0
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF l2 f) l2 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a = c
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  unfold adapted_couple in H2, H3; decompose [and] H2; decompose [and] H3;
    clear H3 H2; rewrite H10; rewrite H6; unfold Rmax, Rmin.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2, l3, lf3:Rlist
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
H4:ordered_Rlist l2
H6:pos_Rl l2 0 = Rmin b c
H5:pos_Rl l2 (Init.Nat.pred (Rlength l2)) = Rmax b c
H7:Rlength l2 = S (Rlength lf2)
H9:forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l2 i) (pos_Rl l2 (S i)))
  (pos_Rl lf2 i)
H8:ordered_Rlist l3
H11:pos_Rl l3 0 = Rmin a c
H10:pos_Rl l3 (Init.Nat.pred (Rlength l3)) =
Rmax a c
H12:Rlength l3 = S (Rlength lf3)
H14:forall i : nat,
(i < Init.Nat.pred (Rlength l3))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l3 i) (pos_Rl l3 (S i)))
  (pos_Rl lf3 i)
(if Rle_dec a c then c else a) =
(if Rle_dec b c then b else c)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF l2 f) l2 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a = c
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
      decide (Rle_dec a c) with Hle''; decide (Rle_dec b c) with Hnle';
        reflexivity.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF l2 f) l2 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a = c
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17;
    [ apply StepFun_P21; unfold is_subdivision; split with lf2; apply H2
      | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a = c
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17;
    [ apply StepFun_P21; unfold is_subdivision; split with lf3; apply H3
      | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a < c
Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a = c
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17;
    [ apply (StepFun_P40 H0 H H3 (StepFun_P2 H2)) | apply H1 ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a = c
Int_SF lf1 l1 = Int_SF lf2 l2 + Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf3 l3) with 0.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a = c
Int_SF lf1 l1 = Int_SF lf2 l2 + 0
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a = c
0 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  rewrite Rplus_0_r; eapply StepFun_P17;
    [ apply H1 | apply StepFun_P2; rewrite <- H0 in H2; apply H2 ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hle'':a <= c
H:c < b
H0:a = c
0 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  symmetry ; eapply StepFun_P8; [ apply H3 | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf2 l2) with (Int_SF lf3 l3 + Int_SF lf1 l1).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf1 l1 + - (Int_SF lf3 l3 + Int_SF lf1 l1) =
- Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  ring.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  elim Hle; intro.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf2 l2) with
  (Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1)).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF lf3 l3 + Int_SF lf1 l1 =
Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF l3 f) l3 + Int_SF lf1 l1 =
Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF l3 f) l3 + Int_SF (FF l1 f) l1 =
Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  symmetry ; apply StepFun_P42.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
pos_Rl l3 (Init.Nat.pred (Rlength l3)) = pos_Rl l1 0
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  unfold adapted_couple in H1, H3; decompose [and] H1; decompose [and] H3;
    clear H3 H1; rewrite H9; rewrite H5; unfold Rmax, Rmin;
      decide (Rle_dec a c) with Hnle''; decide (Rle_dec a b) with Hle; reflexivity.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17;
    [ apply StepFun_P21; unfold is_subdivision; split with lf1; apply H1
      | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17;
    [ apply StepFun_P21; unfold is_subdivision; split with lf3; apply H3
      | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
adapted_couple ?f ?a ?b (cons_Rlist l3 l1)
  (FF (cons_Rlist l3 l1) f)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
adapted_couple ?f ?a ?b l2 lf2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  assert (H0 : c < a).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
c < a
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
H0:c < a
adapted_couple ?f ?a ?b (cons_Rlist l3 l1)
  (FF (cons_Rlist l3 l1) f)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
adapted_couple ?f ?a ?b l2 lf2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  auto with real.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
H0:c < a
adapted_couple ?f ?a ?b (cons_Rlist l3 l1)
  (FF (cons_Rlist l3 l1) f)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
adapted_couple ?f ?a ?b l2 lf2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply (StepFun_P40 H0 H (StepFun_P2 H3) H1).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a < b
adapted_couple f c b l2 lf2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply StepFun_P2; apply H2.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf1 l1) with 0.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
Int_SF lf3 l3 + 0 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
0 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  rewrite Rplus_0_r; eapply StepFun_P17;
    [ apply H3 | rewrite <- H in H2; apply H2 ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hle:a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:a = b
0 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  symmetry ; eapply StepFun_P8; [ apply H1 | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  assert (H : b < a).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
b < a
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  auto with real.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
- Int_SF lf1 l1 + Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf2 l2) with (Int_SF lf3 l3 + Int_SF lf1 l1).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
- Int_SF lf1 l1 + (Int_SF lf3 l3 + Int_SF lf1 l1) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  ring.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
Int_SF lf3 l3 + Int_SF lf1 l1 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  rewrite Rplus_comm; elim Hle''; intro.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF lf1 l1 + Int_SF lf3 l3 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
Int_SF lf1 l1 + Int_SF lf3 l3 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf2 l2) with
  (Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3)).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF lf1 l1 + Int_SF lf3 l3 =
Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
Int_SF lf1 l1 + Int_SF lf3 l3 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF lf1 l1 + Int_SF (FF l3 f) l3 =
Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
Int_SF lf1 l1 + Int_SF lf3 l3 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF l1 f) l1 + Int_SF (FF l3 f) l3 =
Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
Int_SF lf1 l1 + Int_SF lf3 l3 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  symmetry ; apply StepFun_P42.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
pos_Rl l1 (Init.Nat.pred (Rlength l1)) = pos_Rl l3 0
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
Int_SF lf1 l1 + Int_SF lf3 l3 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  unfold adapted_couple in H1, H3; decompose [and] H1; decompose [and] H3;
    clear H3 H1; rewrite H11; rewrite H5; unfold Rmax, Rmin;
      decide (Rle_dec a c) with Hle''; decide (Rle_dec a b) with Hnle;
        reflexivity.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
Int_SF lf1 l1 + Int_SF lf3 l3 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17;
    [ apply StepFun_P21; unfold is_subdivision; split with lf1; apply H1
      | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
Int_SF lf1 l1 + Int_SF lf3 l3 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17;
    [ apply StepFun_P21; unfold is_subdivision; split with lf3; apply H3
      | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3) =
Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
Int_SF lf1 l1 + Int_SF lf3 l3 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
adapted_couple ?f ?a ?b (cons_Rlist l1 l3)
  (FF (cons_Rlist l1 l3) f)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
adapted_couple ?f ?a ?b l2 lf2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
Int_SF lf1 l1 + Int_SF lf3 l3 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply (StepFun_P40 H H0 (StepFun_P2 H1) H3).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a < c
adapted_couple f b c l2 lf2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
Int_SF lf1 l1 + Int_SF lf3 l3 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply H2.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
Int_SF lf1 l1 + Int_SF lf3 l3 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf3 l3) with 0.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
Int_SF lf1 l1 + 0 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
0 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  rewrite Rplus_0_r; eapply StepFun_P17;
    [ apply H1 | rewrite <- H0 in H2; apply StepFun_P2; apply H2 ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hle'':a <= c
H:b < a
H0:a = c
0 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  symmetry ; eapply StepFun_P8; [ apply H3 | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  assert (H : c < a).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
c < a
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  auto with real.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
- Int_SF lf1 l1 + Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf1 l1) with (Int_SF lf2 l2 + Int_SF lf3 l3).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
- (Int_SF lf2 l2 + Int_SF lf3 l3) + Int_SF lf2 l2 =
- Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  ring.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  elim Hle'; intro.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf1 l1) with
  (Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3)).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF lf2 l2 + Int_SF lf3 l3 =
Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF lf2 l2 + Int_SF (FF l3 f) l3 =
Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF l2 f) l2 + Int_SF (FF l3 f) l3 =
Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF l2 f) l2 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  symmetry ; apply StepFun_P42.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
pos_Rl l2 (Init.Nat.pred (Rlength l2)) = pos_Rl l3 0
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF l2 f) l2 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  unfold adapted_couple in H2, H3; decompose [and] H2; decompose [and] H3;
    clear H3 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin;
      decide (Rle_dec a c) with Hnle''; decide (Rle_dec b c) with Hle';
        reflexivity.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF l2 f) l2 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17;
    [ apply StepFun_P21; unfold is_subdivision; split with lf2; apply H2
      | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF l3 f) l3 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17;
    [ apply StepFun_P21; unfold is_subdivision; split with lf3; apply H3
      | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3) =
Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
adapted_couple ?f ?a ?b (cons_Rlist l2 l3)
  (FF (cons_Rlist l2 l3) f)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
adapted_couple ?f ?a ?b l1 lf1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply (StepFun_P40 H0 H H2 (StepFun_P2 H3)).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b < c
adapted_couple f b a l1 lf1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply StepFun_P2; apply H1.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
Int_SF lf2 l2 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf2 l2) with 0.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
0 + Int_SF lf3 l3 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
0 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  rewrite Rplus_0_l; eapply StepFun_P17;
    [ apply H3 | rewrite H0 in H1; apply H1 ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hle':b <= c
Hnle'':~ a <= c
H:c < a
H0:b = c
0 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  symmetry; eapply StepFun_P8; [ apply H2 | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  elim Hnle'; apply Rle_trans with a; try assumption.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hle'':a <= c
b <= a
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  auto with real.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  assert (H : c < b).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
c < b
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  auto with real.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  assert (H0 : b < a).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
b < a
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  auto with real.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
- Int_SF lf1 l1 + - Int_SF lf2 l2 = - Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf3 l3) with (Int_SF lf2 l2 + Int_SF lf1 l1).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
- Int_SF lf1 l1 + - Int_SF lf2 l2 =
- (Int_SF lf2 l2 + Int_SF lf1 l1)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF lf2 l2 + Int_SF lf1 l1 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  ring.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF lf2 l2 + Int_SF lf1 l1 = Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf3 l3) with
  (Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1)).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF lf2 l2 + Int_SF lf1 l1 =
Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF lf2 l2 + Int_SF (FF l1 f) l1 =
Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF l2 f) l2 + Int_SF (FF l1 f) l1 =
Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF l2 f) l2 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  symmetry ; apply StepFun_P42.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
pos_Rl l2 (Init.Nat.pred (Rlength l2)) = pos_Rl l1 0
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF l2 f) l2 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  unfold adapted_couple in H2, H1; decompose [and] H2; decompose [and] H1;
    clear H1 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin;
      decide (Rle_dec a b) with Hnle; decide (Rle_dec b c) with Hnle';
         reflexivity.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF l2 f) l2 = Int_SF lf2 l2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17;
    [ apply StepFun_P21; unfold is_subdivision; split with lf2; apply H2
      | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF l1 f) l1 = Int_SF lf1 l1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17;
    [ apply StepFun_P21; unfold is_subdivision; split with lf1; apply H1
      | assumption ].f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1) =
Int_SF lf3 l3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
adapted_couple ?f ?a ?b (cons_Rlist l2 l1)
  (FF (cons_Rlist l2 l1) f)
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
adapted_couple ?f ?a ?b l3 lf3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply (StepFun_P40 H H0 (StepFun_P2 H2) (StepFun_P2 H1)).f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
Hnle:~ a <= b
Hnle':~ b <= c
Hnle'':~ a <= c
H:c < b
H0:b < a
adapted_couple f c a l3 lf3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply StepFun_P2; apply H3.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a c
 then Int_SF lf3 l3
 else - Int_SF lf3 l3) =
RiemannInt_SF {| fe := f; pre := pr3 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  unfold RiemannInt_SF; case (Rle_dec a c); intro.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
r:a <= c
Int_SF lf3 l3 =
Int_SF (subdivision_val {| fe := f; pre := pr3 |})
  (subdivision {| fe := f; pre := pr3 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= c
- Int_SF lf3 l3 =
-
Int_SF (subdivision_val {| fe := f; pre := pr3 |})
  (subdivision {| fe := f; pre := pr3 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
r:a <= c
adapted_couple ?f ?a ?b l3 lf3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
r:a <= c
adapted_couple ?f ?a ?b
  (subdivision {| fe := f; pre := pr3 |})
  (subdivision_val {| fe := f; pre := pr3 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= c
- Int_SF lf3 l3 =
-
Int_SF (subdivision_val {| fe := f; pre := pr3 |})
  (subdivision {| fe := f; pre := pr3 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply H3.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
r:a <= c
adapted_couple f a c
  (subdivision {| fe := f; pre := pr3 |})
  (subdivision_val {| fe := f; pre := pr3 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= c
- Int_SF lf3 l3 =
-
Int_SF (subdivision_val {| fe := f; pre := pr3 |})
  (subdivision {| fe := f; pre := pr3 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  change
    (adapted_couple (mkStepFun pr3) a c (subdivision (mkStepFun pr3))
      (subdivision_val (mkStepFun pr3))); apply StepFun_P1.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= c
- Int_SF lf3 l3 =
-
Int_SF (subdivision_val {| fe := f; pre := pr3 |})
  (subdivision {| fe := f; pre := pr3 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply Ropp_eq_compat; eapply StepFun_P17.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= c
adapted_couple ?f ?a ?b l3 lf3
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= c
adapted_couple ?f ?a ?b
  (subdivision {| fe := f; pre := pr3 |})
  (subdivision_val {| fe := f; pre := pr3 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply H3.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= c
adapted_couple f a c
  (subdivision {| fe := f; pre := pr3 |})
  (subdivision_val {| fe := f; pre := pr3 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  change
    (adapted_couple (mkStepFun pr3) a c (subdivision (mkStepFun pr3))
      (subdivision_val (mkStepFun pr3))); apply StepFun_P1.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec b c
 then Int_SF lf2 l2
 else - Int_SF lf2 l2) =
RiemannInt_SF {| fe := f; pre := pr2 |}
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  unfold RiemannInt_SF; case (Rle_dec b c); intro.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
r:b <= c
Int_SF lf2 l2 =
Int_SF (subdivision_val {| fe := f; pre := pr2 |})
  (subdivision {| fe := f; pre := pr2 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ b <= c
- Int_SF lf2 l2 =
-
Int_SF (subdivision_val {| fe := f; pre := pr2 |})
  (subdivision {| fe := f; pre := pr2 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  eapply StepFun_P17.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
r:b <= c
adapted_couple ?f ?a ?b l2 lf2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
r:b <= c
adapted_couple ?f ?a ?b
  (subdivision {| fe := f; pre := pr2 |})
  (subdivision_val {| fe := f; pre := pr2 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ b <= c
- Int_SF lf2 l2 =
-
Int_SF (subdivision_val {| fe := f; pre := pr2 |})
  (subdivision {| fe := f; pre := pr2 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply H2.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
r:b <= c
adapted_couple f b c
  (subdivision {| fe := f; pre := pr2 |})
  (subdivision_val {| fe := f; pre := pr2 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ b <= c
- Int_SF lf2 l2 =
-
Int_SF (subdivision_val {| fe := f; pre := pr2 |})
  (subdivision {| fe := f; pre := pr2 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  change
    (adapted_couple (mkStepFun pr2) b c (subdivision (mkStepFun pr2))
      (subdivision_val (mkStepFun pr2))); apply StepFun_P1.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ b <= c
- Int_SF lf2 l2 =
-
Int_SF (subdivision_val {| fe := f; pre := pr2 |})
  (subdivision {| fe := f; pre := pr2 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply Ropp_eq_compat; eapply StepFun_P17.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ b <= c
adapted_couple ?f ?a ?b l2 lf2
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ b <= c
adapted_couple ?f ?a ?b
  (subdivision {| fe := f; pre := pr2 |})
  (subdivision_val {| fe := f; pre := pr2 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  apply H2.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ b <= c
adapted_couple f b c
  (subdivision {| fe := f; pre := pr2 |})
  (subdivision_val {| fe := f; pre := pr2 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  change
    (adapted_couple (mkStepFun pr2) b c (subdivision (mkStepFun pr2))
      (subdivision_val (mkStepFun pr2))); apply StepFun_P1.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
(if Rle_dec a b
 then Int_SF lf1 l1
 else - Int_SF lf1 l1) =
RiemannInt_SF {| fe := f; pre := pr1 |}
  unfold RiemannInt_SF; case (Rle_dec a b); intro.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
r:a <= b
Int_SF lf1 l1 =
Int_SF (subdivision_val {| fe := f; pre := pr1 |})
  (subdivision {| fe := f; pre := pr1 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= b
- Int_SF lf1 l1 =
-
Int_SF (subdivision_val {| fe := f; pre := pr1 |})
  (subdivision {| fe := f; pre := pr1 |})
  eapply StepFun_P17.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
r:a <= b
adapted_couple ?f ?a ?b l1 lf1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
r:a <= b
adapted_couple ?f ?a ?b
  (subdivision {| fe := f; pre := pr1 |})
  (subdivision_val {| fe := f; pre := pr1 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= b
- Int_SF lf1 l1 =
-
Int_SF (subdivision_val {| fe := f; pre := pr1 |})
  (subdivision {| fe := f; pre := pr1 |})
  apply H1.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
r:a <= b
adapted_couple f a b
  (subdivision {| fe := f; pre := pr1 |})
  (subdivision_val {| fe := f; pre := pr1 |})
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= b
- Int_SF lf1 l1 =
-
Int_SF (subdivision_val {| fe := f; pre := pr1 |})
  (subdivision {| fe := f; pre := pr1 |})
  change
    (adapted_couple (mkStepFun pr1) a b (subdivision (mkStepFun pr1))
      (subdivision_val (mkStepFun pr1))); apply StepFun_P1.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= b
- Int_SF lf1 l1 =
-
Int_SF (subdivision_val {| fe := f; pre := pr1 |})
  (subdivision {| fe := f; pre := pr1 |})
  apply Ropp_eq_compat; eapply StepFun_P17.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= b
adapted_couple ?f ?a ?b l1 lf1
f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= b
adapted_couple ?f ?a ?b
  (subdivision {| fe := f; pre := pr1 |})
  (subdivision_val {| fe := f; pre := pr1 |})
  apply H1.f:R -> R
a, b, c:R
pr1:IsStepFun f a b
pr2:IsStepFun f b c
pr3:IsStepFun f a c
l1, lf1:Rlist
H1:adapted_couple f a b l1 lf1
l2, lf2:Rlist
H2:adapted_couple f b c l2 lf2
l3, lf3:Rlist
H3:adapted_couple f a c l3 lf3
n:~ a <= b
adapted_couple f a b
  (subdivision {| fe := f; pre := pr1 |})
  (subdivision_val {| fe := f; pre := pr1 |})
  change
    (adapted_couple (mkStepFun pr1) a b (subdivision (mkStepFun pr1))
      (subdivision_val (mkStepFun pr1))); apply StepFun_P1.
Qed.

Lemma StepFun_P44 :
  forall (f:R -> R) (a b c:R),
    IsStepFun f a b -> a <= c <= b -> IsStepFun f a c.forall (f : R -> R) (a b c : R),
IsStepFun f a b -> a <= c <= b -> IsStepFun f a c
Proof.forall (f : R -> R) (a b c : R),
IsStepFun f a b -> a <= c <= b -> IsStepFun f a c
  intros f; intros; assert (H0 : a <= b).f:R -> R
a, b, c:R
X:IsStepFun f a b
H:a <= c <= b
a <= b
f:R -> R
a, b, c:R
X:IsStepFun f a b
H:a <= c <= b
H0:a <= b
IsStepFun f a c
  elim H; intros; apply Rle_trans with c; assumption.f:R -> R
a, b, c:R
X:IsStepFun f a b
H:a <= c <= b
H0:a <= b
IsStepFun f a c
  elim H; clear H; intros; unfold IsStepFun in X; unfold is_subdivision in X;
    elim X; clear X; intros l1 [lf1 H2];
      cut
        (forall (l1 lf1:Rlist) (a b c:R) (f:R -> R),
          adapted_couple f a b l1 lf1 ->
          a <= c <= b ->
          { l:Rlist & { l0:Rlist & adapted_couple f a c l l0 } }).f:R -> R
a, b, c:R
H0:a <= b
H:a <= c
H1:c <= b
l1, lf1:Rlist
H2:adapted_couple f a b l1 lf1
(forall (l2 lf2 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
 adapted_couple f0 a0 b0 l2 lf2 ->
 a0 <= c0 <= b0 ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f0 a0 c0 l l0}}) ->
IsStepFun f a c
f:R -> R
a, b, c:R
H0:a <= b
H:a <= c
H1:c <= b
l1, lf1:Rlist
H2:adapted_couple f a b l1 lf1
forall (l0 lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 l0 lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l2 : Rlist & adapted_couple f0 a0 c0 l l2}}
  intro X; unfold IsStepFun; unfold is_subdivision; eapply X.f:R -> R
a, b, c:R
H0:a <= b
H:a <= c
H1:c <= b
l1, lf1:Rlist
H2:adapted_couple f a b l1 lf1
X:forall (l0 lf0 : Rlist) (a0 b0 c0 : R)
  (f0 : R -> R),
adapted_couple f0 a0 b0 l0 lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l2 : Rlist & adapted_couple f0 a0 c0 l l2}}
adapted_couple f a ?b ?l1 ?lf1
f:R -> R
a, b, c:R
H0:a <= b
H:a <= c
H1:c <= b
l1, lf1:Rlist
H2:adapted_couple f a b l1 lf1
X:forall (l0 lf0 : Rlist) (a0 b0 c0 : R)
  (f0 : R -> R),
adapted_couple f0 a0 b0 l0 lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l2 : Rlist & adapted_couple f0 a0 c0 l l2}}
a <= c <= ?b
f:R -> R
a, b, c:R
H0:a <= b
H:a <= c
H1:c <= b
l1, lf1:Rlist
H2:adapted_couple f a b l1 lf1
forall (l0 lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 l0 lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l2 : Rlist & adapted_couple f0 a0 c0 l l2}}
  apply H2.f:R -> R
a, b, c:R
H0:a <= b
H:a <= c
H1:c <= b
l1, lf1:Rlist
H2:adapted_couple f a b l1 lf1
X:forall (l0 lf0 : Rlist) (a0 b0 c0 : R)
  (f0 : R -> R),
adapted_couple f0 a0 b0 l0 lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l2 : Rlist & adapted_couple f0 a0 c0 l l2}}
a <= c <= b
f:R -> R
a, b, c:R
H0:a <= b
H:a <= c
H1:c <= b
l1, lf1:Rlist
H2:adapted_couple f a b l1 lf1
forall (l0 lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 l0 lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l2 : Rlist & adapted_couple f0 a0 c0 l l2}}
  split; assumption.f:R -> R
a, b, c:R
H0:a <= b
H:a <= c
H1:c <= b
l1, lf1:Rlist
H2:adapted_couple f a b l1 lf1
forall (l0 lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 l0 lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l2 : Rlist & adapted_couple f0 a0 c0 l l2}}
  clear f a b c H0 H H1 H2 l1 lf1; simple induction l1.l1:Rlist
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b nil lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
forall (r : R) (r0 : Rlist),
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b r0 lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r r0) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  intros; unfold adapted_couple in H; decompose [and] H; clear H; simpl in H4;
    discriminate.l1:Rlist
forall (r : R) (r0 : Rlist),
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b r0 lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r r0) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  simple induction r0.l1:Rlist
r:R
r0:Rlist
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b nil lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r nil) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f a c l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  intros X lf1 a b c f H H0; assert (H1 : a = b).l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
a = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f a c l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  unfold adapted_couple in H; decompose [and] H; clear H; simpl in H3;
    simpl in H2; assert (H7 : a <= b).l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
a <= b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
a = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f a c l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  elim H0; intros; apply Rle_trans with c; assumption.l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
a = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f a c l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  replace a with (Rmin a b).l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmin a b = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmin a b = a
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f a c l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  pattern b at 2; replace b with (Rmax a b).l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmin a b = Rmax a b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmax a b = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmin a b = a
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f a c l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  rewrite <- H2; rewrite H3; reflexivity.l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmax a b = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmin a b = a
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f a c l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  unfold Rmax; decide (Rle_dec a b) with H7; reflexivity.l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmin a b = a
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f a c l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  unfold Rmin; decide (Rle_dec a b) with H7; reflexivity.l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f a c l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  split with (cons r nil); split with lf1; assert (H2 : c = b).l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
c = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
H2:c = b
adapted_couple f a c (cons r nil) lf1
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f a c l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  rewrite H1 in H0; elim H0; intros; apply Rle_antisym; assumption.l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
H2:c = b
adapted_couple f a c (cons r nil) lf1
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f a c l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  rewrite H2; assumption.l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f a c l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f a c l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  intros r1 r2 _ X0 lf1 a b c f H H0; induction  lf1 as [| r3 lf1 Hreclf1].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf1 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 (cons r1 r2) lf1 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2)) nil
H0:a <= c <= b
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 (cons r1 r2) lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f a c l l0}}
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  unfold adapted_couple in H; decompose [and] H; clear H; simpl in H4;
    discriminate.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 (cons r1 r2) lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f a c l l0}}
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  clear Hreclf1; assert (H1 : {c <= r1} + {r1 < c}).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 (cons r1 r2) lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
{c <= r1} + {r1 < c}
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 (cons r1 r2) lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  case (Rle_dec c r1); intro; [ left; assumption | right; auto with real ].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 (cons r1 r2) lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a0 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  elim H1; intro a0.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  split with (cons r (cons c nil)); split with (cons r3 nil);
    unfold adapted_couple in H; decompose [and] H; clear H;
      assert (H6 : r = a).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
r = a
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H6:r = a
adapted_couple f a c (cons r (cons c nil))
  (cons r3 nil)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  simpl in H4; rewrite H4; unfold Rmin; case (Rle_dec a b) as [|[]];
    [ reflexivity
      | elim H0; intros; apply Rle_trans with c; assumption ].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H6:r = a
adapted_couple f a c (cons r (cons c nil))
  (cons r3 nil)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  elim H0; clear H0; intros; unfold adapted_couple; repeat split.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H6:r = a
H:a <= c
H0:c <= b
ordered_Rlist (cons r (cons c nil))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H6:r = a
H:a <= c
H0:c <= b
pos_Rl (cons r (cons c nil)) 0 = Rmin a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H6:r = a
H:a <= c
H0:c <= b
pos_Rl (cons r (cons c nil))
  (Init.Nat.pred (Rlength (cons r (cons c nil)))) =
Rmax a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H6:r = a
H:a <= c
H0:c <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons c nil))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons c nil)) i)
     (pos_Rl (cons r (cons c nil)) (S i)))
  (pos_Rl (cons r3 nil) i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  rewrite H6; unfold ordered_Rlist; intros; simpl in H8; inversion H8;
    [ simpl; assumption | elim (le_Sn_O _ H10) ].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H6:r = a
H:a <= c
H0:c <= b
pos_Rl (cons r (cons c nil)) 0 = Rmin a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H6:r = a
H:a <= c
H0:c <= b
pos_Rl (cons r (cons c nil))
  (Init.Nat.pred (Rlength (cons r (cons c nil)))) =
Rmax a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H6:r = a
H:a <= c
H0:c <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons c nil))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons c nil)) i)
     (pos_Rl (cons r (cons c nil)) (S i)))
  (pos_Rl (cons r3 nil) i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  simpl; unfold Rmin; decide (Rle_dec a c) with H; assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H6:r = a
H:a <= c
H0:c <= b
pos_Rl (cons r (cons c nil))
  (Init.Nat.pred (Rlength (cons r (cons c nil)))) =
Rmax a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H6:r = a
H:a <= c
H0:c <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons c nil))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons c nil)) i)
     (pos_Rl (cons r (cons c nil)) (S i)))
  (pos_Rl (cons r3 nil) i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  simpl; unfold Rmax; decide (Rle_dec a c) with H; reflexivity.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H6:r = a
H:a <= c
H0:c <= b
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons c nil))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons c nil)) i)
     (pos_Rl (cons r (cons c nil)) (S i)))
  (pos_Rl (cons r3 nil) i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  unfold constant_D_eq, open_interval; intros; simpl in H8;
    inversion H8.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H6:r = a
H:a <= c
H0:c <= b
i:nat
H8:(i < 1)%nat
x:R
H9:pos_Rl (cons r (cons c nil)) i < x <
pos_Rl (cons r (cons c nil)) (S i)
H11:i = 0%nat
f x = pos_Rl (cons r3 nil) 0
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H6:r = a
H:a <= c
H0:c <= b
i:nat
H8:(i < 1)%nat
x:R
H9:pos_Rl (cons r (cons c nil)) i < x <
pos_Rl (cons r (cons c nil)) (S i)
m:nat
H11:(S i <= 0)%nat
H10:m = 0%nat
f x = pos_Rl (cons r3 nil) i
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  simpl; assert (H10 := H7 0%nat);
    assert (H12 : (0 < pred (Rlength (cons r (cons r1 r2))))%nat).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H6:r = a
H:a <= c
H0:c <= b
i:nat
H8:(i < 1)%nat
x:R
H9:pos_Rl (cons r (cons c nil)) i < x <
pos_Rl (cons r (cons c nil)) (S i)
H11:i = 0%nat
H10:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) 0)
     (pos_Rl (cons r (cons r1 r2)) 1))
  (pos_Rl (cons r3 lf1) 0)
(0 < Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H6:r = a
H:a <= c
H0:c <= b
i:nat
H8:(i < 1)%nat
x:R
H9:pos_Rl (cons r (cons c nil)) i < x <
pos_Rl (cons r (cons c nil)) (S i)
H11:i = 0%nat
H10:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) 0)
     (pos_Rl (cons r (cons r1 r2)) 1))
  (pos_Rl (cons r3 lf1) 0)
H12:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
f x = r3
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H6:r = a
H:a <= c
H0:c <= b
i:nat
H8:(i < 1)%nat
x:R
H9:pos_Rl (cons r (cons c nil)) i < x <
pos_Rl (cons r (cons c nil)) (S i)
m:nat
H11:(S i <= 0)%nat
H10:m = 0%nat
f x = pos_Rl (cons r3 nil) i
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  simpl; apply lt_O_Sn.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H6:r = a
H:a <= c
H0:c <= b
i:nat
H8:(i < 1)%nat
x:R
H9:pos_Rl (cons r (cons c nil)) i < x <
pos_Rl (cons r (cons c nil)) (S i)
H11:i = 0%nat
H10:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) 0)
     (pos_Rl (cons r (cons r1 r2)) 1))
  (pos_Rl (cons r3 lf1) 0)
H12:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
f x = r3
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H6:r = a
H:a <= c
H0:c <= b
i:nat
H8:(i < 1)%nat
x:R
H9:pos_Rl (cons r (cons c nil)) i < x <
pos_Rl (cons r (cons c nil)) (S i)
m:nat
H11:(S i <= 0)%nat
H10:m = 0%nat
f x = pos_Rl (cons r3 nil) i
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  apply (H10 H12); unfold open_interval; simpl;
    rewrite H11 in H9; simpl in H9; elim H9; clear H9;
      intros; split; try assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H6:r = a
H:a <= c
H0:c <= b
i:nat
H8:(i < 1)%nat
x:R
H11:i = 0%nat
H10:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) 0)
     (pos_Rl (cons r (cons r1 r2)) 1))
  (pos_Rl (cons r3 lf1) 0)
H12:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
H9:r < x
H13:x < c
x < r1
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H6:r = a
H:a <= c
H0:c <= b
i:nat
H8:(i < 1)%nat
x:R
H9:pos_Rl (cons r (cons c nil)) i < x <
pos_Rl (cons r (cons c nil)) (S i)
m:nat
H11:(S i <= 0)%nat
H10:m = 0%nat
f x = pos_Rl (cons r3 nil) i
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  apply Rlt_le_trans with c; assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H6:r = a
H:a <= c
H0:c <= b
i:nat
H8:(i < 1)%nat
x:R
H9:pos_Rl (cons r (cons c nil)) i < x <
pos_Rl (cons r (cons c nil)) (S i)
m:nat
H11:(S i <= 0)%nat
H10:m = 0%nat
f x = pos_Rl (cons r3 nil) i
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  elim (le_Sn_O _ H11).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
  cut (adapted_couple f r1 b (cons r1 r2) lf1).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1 ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  cut (r1 <= c <= b).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b ->
adapted_couple f r1 b (cons r1 r2) lf1 ->
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  intros.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
{l : Rlist & {l0 : Rlist & adapted_couple f a c l l0}}
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  elim (X0 _ _ _ _ _ H3 H2); intros l1' [lf1' H4]; split with (cons r l1');
    split with (cons r3 lf1'); unfold adapted_couple in H, H4;
      decompose [and] H; decompose [and] H4; clear H H4 X0;
        assert (H14 : a <= b).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist l1'
H12:pos_Rl l1' 0 = Rmin r1 c
H11:pos_Rl l1' (Init.Nat.pred (Rlength l1')) =
Rmax r1 c
H13:Rlength l1' = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength l1'))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1' i)
     (pos_Rl l1' (S i))) 
  (pos_Rl lf1' i)
a <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist l1'
H12:pos_Rl l1' 0 = Rmin r1 c
H11:pos_Rl l1' (Init.Nat.pred (Rlength l1')) =
Rmax r1 c
H13:Rlength l1' = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength l1'))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1' i)
     (pos_Rl l1' (S i))) 
  (pos_Rl lf1' i)
H14:a <= b
adapted_couple f a c (cons r l1') (cons r3 lf1')
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  elim H0; intros; apply Rle_trans with c; assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist l1'
H12:pos_Rl l1' 0 = Rmin r1 c
H11:pos_Rl l1' (Init.Nat.pred (Rlength l1')) =
Rmax r1 c
H13:Rlength l1' = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength l1'))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1' i)
     (pos_Rl l1' (S i))) 
  (pos_Rl lf1' i)
H14:a <= b
adapted_couple f a c (cons r l1') (cons r3 lf1')
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  assert (H16 : r = a).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist l1'
H12:pos_Rl l1' 0 = Rmin r1 c
H11:pos_Rl l1' (Init.Nat.pred (Rlength l1')) =
Rmax r1 c
H13:Rlength l1' = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength l1'))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1' i)
     (pos_Rl l1' (S i))) 
  (pos_Rl lf1' i)
H14:a <= b
r = a
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist l1'
H12:pos_Rl l1' 0 = Rmin r1 c
H11:pos_Rl l1' (Init.Nat.pred (Rlength l1')) =
Rmax r1 c
H13:Rlength l1' = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength l1'))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1' i)
     (pos_Rl l1' (S i))) 
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
adapted_couple f a c (cons r l1') (cons r3 lf1')
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  simpl in H7; rewrite H7; unfold Rmin; decide (Rle_dec a b) with H14;
    reflexivity.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist l1'
H12:pos_Rl l1' 0 = Rmin r1 c
H11:pos_Rl l1' (Init.Nat.pred (Rlength l1')) =
Rmax r1 c
H13:Rlength l1' = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength l1'))%nat ->
constant_D_eq f
  (open_interval (pos_Rl l1' i)
     (pos_Rl l1' (S i))) 
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
adapted_couple f a c (cons r l1') (cons r3 lf1')
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  induction  l1' as [| r4 l1' Hrecl1'].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist nil
H12:pos_Rl nil 0 = Rmin r1 c
H11:pos_Rl nil (Init.Nat.pred (Rlength nil)) =
Rmax r1 c
H13:Rlength nil = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength nil))%nat ->
constant_D_eq f
  (open_interval (pos_Rl nil i)
     (pos_Rl nil (S i))) 
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
adapted_couple f a c (cons r nil) (cons r3 lf1')
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Hrecl1':ordered_Rlist l1' ->
pos_Rl l1' 0 = Rmin r1 c ->
pos_Rl l1' (Init.Nat.pred (Rlength l1')) =
Rmax r1 c ->
Rlength l1' = S (Rlength lf1') ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength l1'))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l1' i)
      (pos_Rl l1' (S i))) 
   (pos_Rl lf1' i)) ->
adapted_couple f a c (cons r l1')
  (cons r3 lf1')
adapted_couple f a c (cons r (cons r4 l1'))
  (cons r3 lf1')
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  simpl in H13; discriminate.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Hrecl1':ordered_Rlist l1' ->
pos_Rl l1' 0 = Rmin r1 c ->
pos_Rl l1' (Init.Nat.pred (Rlength l1')) =
Rmax r1 c ->
Rlength l1' = S (Rlength lf1') ->
(forall i : nat,
 (i < Init.Nat.pred (Rlength l1'))%nat ->
 constant_D_eq f
   (open_interval (pos_Rl l1' i)
      (pos_Rl l1' (S i))) 
   (pos_Rl lf1' i)) ->
adapted_couple f a c (cons r l1')
  (cons r3 lf1')
adapted_couple f a c (cons r (cons r4 l1'))
  (cons r3 lf1')
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  clear Hrecl1'; unfold adapted_couple; repeat split.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
ordered_Rlist (cons r (cons r4 l1'))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1')) 0 = Rmin a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1'))
  (Init.Nat.pred (Rlength (cons r (cons r4 l1')))) =
Rmax a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Rlength (cons r (cons r4 l1')) =
S (Rlength (cons r3 lf1'))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r4 l1'))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r4 l1')) i)
     (pos_Rl (cons r (cons r4 l1')) (S i)))
  (pos_Rl (cons r3 lf1') i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  unfold ordered_Rlist; intros; simpl in H; induction  i as [| i Hreci].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
H:(0 < S (Rlength l1'))%nat
pos_Rl (cons r (cons r4 l1')) 0 <=
pos_Rl (cons r (cons r4 l1')) 1
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
Hreci:(i < S (Rlength l1'))%nat ->
pos_Rl (cons r (cons r4 l1')) i <=
pos_Rl (cons r (cons r4 l1')) (S i)
pos_Rl (cons r (cons r4 l1')) (S i) <=
pos_Rl (cons r (cons r4 l1')) (S (S i))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1')) 0 = Rmin a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1'))
  (Init.Nat.pred (Rlength (cons r (cons r4 l1')))) =
Rmax a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Rlength (cons r (cons r4 l1')) =
S (Rlength (cons r3 lf1'))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r4 l1'))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r4 l1')) i)
     (pos_Rl (cons r (cons r4 l1')) (S i)))
  (pos_Rl (cons r3 lf1') i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  simpl; replace r4 with r1.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
H:(0 < S (Rlength l1'))%nat
r <= r1
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
H:(0 < S (Rlength l1'))%nat
r1 = r4
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
Hreci:(i < S (Rlength l1'))%nat ->
pos_Rl (cons r (cons r4 l1')) i <=
pos_Rl (cons r (cons r4 l1')) (S i)
pos_Rl (cons r (cons r4 l1')) (S i) <=
pos_Rl (cons r (cons r4 l1')) (S (S i))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1')) 0 = Rmin a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1'))
  (Init.Nat.pred (Rlength (cons r (cons r4 l1')))) =
Rmax a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Rlength (cons r (cons r4 l1')) =
S (Rlength (cons r3 lf1'))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r4 l1'))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r4 l1')) i)
     (pos_Rl (cons r (cons r4 l1')) (S i)))
  (pos_Rl (cons r3 lf1') i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  apply (H5 0%nat).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
H:(0 < S (Rlength l1'))%nat
(0 < Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
H:(0 < S (Rlength l1'))%nat
r1 = r4
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
Hreci:(i < S (Rlength l1'))%nat ->
pos_Rl (cons r (cons r4 l1')) i <=
pos_Rl (cons r (cons r4 l1')) (S i)
pos_Rl (cons r (cons r4 l1')) (S i) <=
pos_Rl (cons r (cons r4 l1')) (S (S i))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1')) 0 = Rmin a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1'))
  (Init.Nat.pred (Rlength (cons r (cons r4 l1')))) =
Rmax a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Rlength (cons r (cons r4 l1')) =
S (Rlength (cons r3 lf1'))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r4 l1'))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r4 l1')) i)
     (pos_Rl (cons r (cons r4 l1')) (S i)))
  (pos_Rl (cons r3 lf1') i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  simpl; apply lt_O_Sn.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
H:(0 < S (Rlength l1'))%nat
r1 = r4
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
Hreci:(i < S (Rlength l1'))%nat ->
pos_Rl (cons r (cons r4 l1')) i <=
pos_Rl (cons r (cons r4 l1')) (S i)
pos_Rl (cons r (cons r4 l1')) (S i) <=
pos_Rl (cons r (cons r4 l1')) (S (S i))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1')) 0 = Rmin a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1'))
  (Init.Nat.pred (Rlength (cons r (cons r4 l1')))) =
Rmax a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Rlength (cons r (cons r4 l1')) =
S (Rlength (cons r3 lf1'))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r4 l1'))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r4 l1')) i)
     (pos_Rl (cons r (cons r4 l1')) (S i)))
  (pos_Rl (cons r3 lf1') i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  simpl in H12; rewrite H12; unfold Rmin; case (Rle_dec r1 c) as [|[]];
    [ reflexivity | left; assumption ].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
Hreci:(i < S (Rlength l1'))%nat ->
pos_Rl (cons r (cons r4 l1')) i <=
pos_Rl (cons r (cons r4 l1')) (S i)
pos_Rl (cons r (cons r4 l1')) (S i) <=
pos_Rl (cons r (cons r4 l1')) (S (S i))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1')) 0 = Rmin a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1'))
  (Init.Nat.pred (Rlength (cons r (cons r4 l1')))) =
Rmax a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Rlength (cons r (cons r4 l1')) =
S (Rlength (cons r3 lf1'))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r4 l1'))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r4 l1')) i)
     (pos_Rl (cons r (cons r4 l1')) (S i)))
  (pos_Rl (cons r3 lf1') i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  apply (H9 i); simpl; apply lt_S_n; assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1')) 0 = Rmin a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1'))
  (Init.Nat.pred (Rlength (cons r (cons r4 l1')))) =
Rmax a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Rlength (cons r (cons r4 l1')) =
S (Rlength (cons r3 lf1'))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r4 l1'))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r4 l1')) i)
     (pos_Rl (cons r (cons r4 l1')) (S i)))
  (pos_Rl (cons r3 lf1') i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  simpl; unfold Rmin; case (Rle_dec a c) as [|[]];
    [ assumption | elim H0; intros; assumption ].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1'))
  (Init.Nat.pred (Rlength (cons r (cons r4 l1')))) =
Rmax a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Rlength (cons r (cons r4 l1')) =
S (Rlength (cons r3 lf1'))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r4 l1'))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r4 l1')) i)
     (pos_Rl (cons r (cons r4 l1')) (S i)))
  (pos_Rl (cons r3 lf1') i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  replace (Rmax a c) with (Rmax r1 c).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
pos_Rl (cons r (cons r4 l1'))
  (Init.Nat.pred (Rlength (cons r (cons r4 l1')))) =
Rmax r1 c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Rmax r1 c = Rmax a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Rlength (cons r (cons r4 l1')) =
S (Rlength (cons r3 lf1'))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r4 l1'))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r4 l1')) i)
     (pos_Rl (cons r (cons r4 l1')) (S i)))
  (pos_Rl (cons r3 lf1') i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  rewrite <- H11; reflexivity.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Rmax r1 c = Rmax a c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Rlength (cons r (cons r4 l1')) =
S (Rlength (cons r3 lf1'))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r4 l1'))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r4 l1')) i)
     (pos_Rl (cons r (cons r4 l1')) (S i)))
  (pos_Rl (cons r3 lf1') i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  unfold Rmax; case (Rle_dec a c) as [|[]]; case (Rle_dec r1 c) as [|[]];
      [ reflexivity
      | left; assumption
      | elim H0; intros; assumption
      | left; assumption ].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
Rlength (cons r (cons r4 l1')) =
S (Rlength (cons r3 lf1'))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r4 l1'))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r4 l1')) i)
     (pos_Rl (cons r (cons r4 l1')) (S i)))
  (pos_Rl (cons r3 lf1') i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  simpl; simpl in H13; rewrite H13; reflexivity.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
forall i : nat,
(i < Init.Nat.pred (Rlength (cons r (cons r4 l1'))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r4 l1')) i)
     (pos_Rl (cons r (cons r4 l1')) (S i)))
  (pos_Rl (cons r3 lf1') i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  intros; simpl in H; unfold constant_D_eq, open_interval; intros;
    induction  i as [| i Hreci].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
H:(0 < S (Rlength l1'))%nat
x:R
H4:pos_Rl (cons r (cons r4 l1')) 0 < x <
pos_Rl (cons r (cons r4 l1')) 1
f x = pos_Rl (cons r3 lf1') 0
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
x:R
H4:pos_Rl (cons r (cons r4 l1')) (S i) < x <
pos_Rl (cons r (cons r4 l1')) (S (S i))
Hreci:(i < S (Rlength l1'))%nat ->
pos_Rl (cons r (cons r4 l1')) i < x <
pos_Rl (cons r (cons r4 l1')) (S i) ->
f x = pos_Rl (cons r3 lf1') i
f x = pos_Rl (cons r3 lf1') (S i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  simpl; assert (H17 := H10 0%nat);
    assert (H18 : (0 < pred (Rlength (cons r (cons r1 r2))))%nat).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
H:(0 < S (Rlength l1'))%nat
x:R
H4:pos_Rl (cons r (cons r4 l1')) 0 < x <
pos_Rl (cons r (cons r4 l1')) 1
H17:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) 0)
     (pos_Rl (cons r (cons r1 r2)) 1))
  (pos_Rl (cons r3 lf1) 0)
(0 < Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
H:(0 < S (Rlength l1'))%nat
x:R
H4:pos_Rl (cons r (cons r4 l1')) 0 < x <
pos_Rl (cons r (cons r4 l1')) 1
H17:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) 0)
     (pos_Rl (cons r (cons r1 r2)) 1))
  (pos_Rl (cons r3 lf1) 0)
H18:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
f x = r3
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
x:R
H4:pos_Rl (cons r (cons r4 l1')) (S i) < x <
pos_Rl (cons r (cons r4 l1')) (S (S i))
Hreci:(i < S (Rlength l1'))%nat ->
pos_Rl (cons r (cons r4 l1')) i < x <
pos_Rl (cons r (cons r4 l1')) (S i) ->
f x = pos_Rl (cons r3 lf1') i
f x = pos_Rl (cons r3 lf1') (S i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  simpl; apply lt_O_Sn.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
H:(0 < S (Rlength l1'))%nat
x:R
H4:pos_Rl (cons r (cons r4 l1')) 0 < x <
pos_Rl (cons r (cons r4 l1')) 1
H17:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) 0)
     (pos_Rl (cons r (cons r1 r2)) 1))
  (pos_Rl (cons r3 lf1) 0)
H18:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
f x = r3
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
x:R
H4:pos_Rl (cons r (cons r4 l1')) (S i) < x <
pos_Rl (cons r (cons r4 l1')) (S (S i))
Hreci:(i < S (Rlength l1'))%nat ->
pos_Rl (cons r (cons r4 l1')) i < x <
pos_Rl (cons r (cons r4 l1')) (S i) ->
f x = pos_Rl (cons r3 lf1') i
f x = pos_Rl (cons r3 lf1') (S i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  apply (H17 H18); unfold open_interval; simpl; simpl in H4;
    elim H4; clear H4; intros; split; try assumption;
      replace r1 with r4.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
H:(0 < S (Rlength l1'))%nat
x:R
H17:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) 0)
     (pos_Rl (cons r (cons r1 r2)) 1))
  (pos_Rl (cons r3 lf1) 0)
H18:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
H4:r < x
H19:x < r4
x < r4
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
H:(0 < S (Rlength l1'))%nat
x:R
H17:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) 0)
     (pos_Rl (cons r (cons r1 r2)) 1))
  (pos_Rl (cons r3 lf1) 0)
H18:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
H4:r < x
H19:x < r4
r4 = r1
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
x:R
H4:pos_Rl (cons r (cons r4 l1')) (S i) < x <
pos_Rl (cons r (cons r4 l1')) (S (S i))
Hreci:(i < S (Rlength l1'))%nat ->
pos_Rl (cons r (cons r4 l1')) i < x <
pos_Rl (cons r (cons r4 l1')) (S i) ->
f x = pos_Rl (cons r3 lf1') i
f x = pos_Rl (cons r3 lf1') (S i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i)
     (pos_Rl (cons r4 l1') (S i)))
  (pos_Rl lf1' i)
H14:a <= b
H16:r = a
H:(0 < S (Rlength l1'))%nat
x:R
H17:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) 0)
     (pos_Rl (cons r (cons r1 r2)) 1))
  (pos_Rl (cons r3 lf1) 0)
H18:(0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
H4:r < x
H19:x < r4
r4 = r1
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
x:R
H4:pos_Rl (cons r (cons r4 l1')) (S i) < x <
pos_Rl (cons r (cons r4 l1')) (S (S i))
Hreci:(i < S (Rlength l1'))%nat ->
pos_Rl (cons r (cons r4 l1')) i < x <
pos_Rl (cons r (cons r4 l1')) (S i) ->
f x = pos_Rl (cons r3 lf1') i
f x = pos_Rl (cons r3 lf1') (S i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  simpl in H12; rewrite H12; unfold Rmin; case (Rle_dec r1 c) as [|[]];
    [ reflexivity | left; assumption ].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
x:R
H4:pos_Rl (cons r (cons r4 l1')) (S i) < x <
pos_Rl (cons r (cons r4 l1')) (S (S i))
Hreci:(i < S (Rlength l1'))%nat ->
pos_Rl (cons r (cons r4 l1')) i < x <
pos_Rl (cons r (cons r4 l1')) (S i) ->
f x = pos_Rl (cons r3 lf1') i
f x = pos_Rl (cons r3 lf1') (S i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  clear Hreci; simpl; apply H15.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
x:R
H4:pos_Rl (cons r (cons r4 l1')) (S i) < x <
pos_Rl (cons r (cons r4 l1')) (S (S i))
(i < Init.Nat.pred (Rlength (cons r4 l1')))%nat
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
x:R
H4:pos_Rl (cons r (cons r4 l1')) (S i) < x <
pos_Rl (cons r (cons r4 l1')) (S (S i))
open_interval (pos_Rl (cons r4 l1') i)
  (pos_Rl (cons r4 l1') (S i)) x
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  simpl; apply lt_S_n; assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
H2:r1 <= c <= b
H3:adapted_couple f r1 b (cons r1 r2) lf1
r4:R
l1', lf1':Rlist
H5:ordered_Rlist (cons r (cons r1 r2))
H7:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H6:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H8:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H10:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
H9:ordered_Rlist (cons r4 l1')
H12:pos_Rl (cons r4 l1') 0 = Rmin r1 c
H11:pos_Rl (cons r4 l1')
  (Init.Nat.pred (Rlength (cons r4 l1'))) =
Rmax r1 c
H13:Rlength (cons r4 l1') = S (Rlength lf1')
H15:forall i0 : nat,
(i0 < Init.Nat.pred (Rlength (cons r4 l1')))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r4 l1') i0)
     (pos_Rl (cons r4 l1') (S i0)))
  (pos_Rl lf1' i0)
H14:a <= b
H16:r = a
i:nat
H:(S i < S (Rlength l1'))%nat
x:R
H4:pos_Rl (cons r (cons r4 l1')) (S i) < x <
pos_Rl (cons r (cons r4 l1')) (S (S i))
open_interval (pos_Rl (cons r4 l1') i)
  (pos_Rl (cons r4 l1') (S i)) x
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  unfold open_interval; apply H4.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  split.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  left; assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  elim H0; intros; assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 a1 c0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  eapply StepFun_P7;
    [ elim H0; intros; apply Rle_trans with c; [ apply H2 | apply H3 ]
      | apply H ].
Qed.

Lemma StepFun_P45 :
  forall (f:R -> R) (a b c:R),
    IsStepFun f a b -> a <= c <= b -> IsStepFun f c b.forall (f : R -> R) (a b c : R),
IsStepFun f a b -> a <= c <= b -> IsStepFun f c b
Proof.forall (f : R -> R) (a b c : R),
IsStepFun f a b -> a <= c <= b -> IsStepFun f c b
  intros f; intros; assert (H0 : a <= b).f:R -> R
a, b, c:R
X:IsStepFun f a b
H:a <= c <= b
a <= b
f:R -> R
a, b, c:R
X:IsStepFun f a b
H:a <= c <= b
H0:a <= b
IsStepFun f c b
  elim H; intros; apply Rle_trans with c; assumption.f:R -> R
a, b, c:R
X:IsStepFun f a b
H:a <= c <= b
H0:a <= b
IsStepFun f c b
  elim H; clear H; intros; unfold IsStepFun in X; unfold is_subdivision in X;
    elim X; clear X; intros l1 [lf1 H2];
      cut
        (forall (l1 lf1:Rlist) (a b c:R) (f:R -> R),
          adapted_couple f a b l1 lf1 ->
          a <= c <= b ->
          { l:Rlist & { l0:Rlist & adapted_couple f c b l l0 } }).f:R -> R
a, b, c:R
H0:a <= b
H:a <= c
H1:c <= b
l1, lf1:Rlist
H2:adapted_couple f a b l1 lf1
(forall (l2 lf2 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
 adapted_couple f0 a0 b0 l2 lf2 ->
 a0 <= c0 <= b0 ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f0 c0 b0 l l0}}) ->
IsStepFun f c b
f:R -> R
a, b, c:R
H0:a <= b
H:a <= c
H1:c <= b
l1, lf1:Rlist
H2:adapted_couple f a b l1 lf1
forall (l0 lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 l0 lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l2 : Rlist & adapted_couple f0 c0 b0 l l2}}
  intro X; unfold IsStepFun; unfold is_subdivision; eapply X;
    [ apply H2 | split; assumption ].f:R -> R
a, b, c:R
H0:a <= b
H:a <= c
H1:c <= b
l1, lf1:Rlist
H2:adapted_couple f a b l1 lf1
forall (l0 lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 l0 lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l2 : Rlist & adapted_couple f0 c0 b0 l l2}}
  clear f a b c H0 H H1 H2 l1 lf1; simple induction l1.l1:Rlist
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b nil lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
forall (r : R) (r0 : Rlist),
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b r0 lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r r0) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  intros; unfold adapted_couple in H; decompose [and] H; clear H; simpl in H4;
    discriminate.l1:Rlist
forall (r : R) (r0 : Rlist),
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b r0 lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r r0) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  simple induction r0.l1:Rlist
r:R
r0:Rlist
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b nil lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r nil) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f c b l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  intros X lf1 a b c f H H0; assert (H1 : a = b).l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
a = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f c b l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  unfold adapted_couple in H; decompose [and] H; clear H; simpl in H3;
    simpl in H2; assert (H7 : a <= b).l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
a <= b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
a = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f c b l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  elim H0; intros; apply Rle_trans with c; assumption.l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
a = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f c b l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  replace a with (Rmin a b).l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmin a b = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmin a b = a
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f c b l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  pattern b at 2; replace b with (Rmax a b).l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmin a b = Rmax a b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmax a b = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmin a b = a
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f c b l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  rewrite <- H2; rewrite H3; reflexivity.l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmax a b = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmin a b = a
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f c b l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  unfold Rmax; decide (Rle_dec a b) with H7; reflexivity.l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:ordered_Rlist (cons r nil)
H3:r = Rmin a b
H2:r = Rmax a b
H4:Rlength (cons r nil) = S (Rlength lf1)
H6:forall i : nat,
(i < Init.Nat.pred (Rlength (cons r nil)))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r nil) i)
     (pos_Rl (cons r nil) (S i))) (pos_Rl lf1 i)
H7:a <= b
Rmin a b = a
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f c b l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  unfold Rmin; decide (Rle_dec a b) with H7; reflexivity.l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f c b l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  split with (cons r nil); split with lf1; assert (H2 : c = b).l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
c = b
l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
H2:c = b
adapted_couple f c b (cons r nil) lf1
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f c b l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  rewrite H1 in H0; elim H0; intros; apply Rle_antisym; assumption.l1:Rlist
r:R
r0:Rlist
X:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 nil lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r nil) lf1
H0:a <= c <= b
H1:a = b
H2:c = b
adapted_couple f c b (cons r nil) lf1
l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f c b l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  rewrite <- H2 in H1; rewrite <- H1; assumption.l1:Rlist
r:R
r0:Rlist
forall (r1 : R) (r2 : Rlist),
((forall (lf1 : Rlist) (a b c : R) (f : R -> R),
  adapted_couple f a b r2 lf1 ->
  a <= c <= b ->
  {l : Rlist &
  {l0 : Rlist & adapted_couple f c b l l0}}) ->
 forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
(forall (lf1 : Rlist) (a b c : R) (f : R -> R),
 adapted_couple f a b (cons r1 r2) lf1 ->
 a <= c <= b ->
 {l : Rlist &
 {l0 : Rlist & adapted_couple f c b l l0}}) ->
forall (lf1 : Rlist) (a b c : R) (f : R -> R),
adapted_couple f a b (cons r (cons r1 r2)) lf1 ->
a <= c <= b ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  intros r1 r2 _ X0 lf1 a b c f H H0; induction  lf1 as [| r3 lf1 Hreclf1].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf1 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 (cons r1 r2) lf1 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2)) nil
H0:a <= c <= b
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 (cons r1 r2) lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f c b l l0}}
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  unfold adapted_couple in H; decompose [and] H; clear H; simpl in H4;
    discriminate.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 (cons r1 r2) lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
Hreclf1:adapted_couple f a b (cons r (cons r1 r2))
  lf1 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f c b l l0}}
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  clear Hreclf1; assert (H1 : {c <= r1} + {r1 < c}).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 (cons r1 r2) lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
{c <= r1} + {r1 < c}
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 (cons r1 r2) lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  case (Rle_dec c r1); intro; [ left; assumption | right; auto with real ].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a0 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a0 b0 (cons r1 r2) lf0 ->
a0 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  elim H1; intro a0.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  split with (cons c (cons r1 r2)); split with (cons r3 lf1);
    unfold adapted_couple in H; decompose [and] H; clear H;
      unfold adapted_couple; repeat split.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
ordered_Rlist (cons c (cons r1 r2))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
pos_Rl (cons c (cons r1 r2)) 0 = Rmin c b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
pos_Rl (cons c (cons r1 r2))
  (Init.Nat.pred (Rlength (cons c (cons r1 r2)))) =
Rmax c b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
Rlength (cons c (cons r1 r2)) =
S (Rlength (cons r3 lf1))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons c (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  unfold ordered_Rlist; intros; simpl in H; induction  i as [| i Hreci].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H:(0 < S (Rlength r2))%nat
pos_Rl (cons c (cons r1 r2)) 0 <=
pos_Rl (cons c (cons r1 r2)) 1
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
i:nat
H:(S i < S (Rlength r2))%nat
Hreci:(i < S (Rlength r2))%nat ->
pos_Rl (cons c (cons r1 r2)) i <=
pos_Rl (cons c (cons r1 r2)) (S i)
pos_Rl (cons c (cons r1 r2)) (S i) <=
pos_Rl (cons c (cons r1 r2)) (S (S i))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
pos_Rl (cons c (cons r1 r2)) 0 = Rmin c b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
pos_Rl (cons c (cons r1 r2))
  (Init.Nat.pred (Rlength (cons c (cons r1 r2)))) =
Rmax c b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
Rlength (cons c (cons r1 r2)) =
S (Rlength (cons r3 lf1))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons c (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  simpl; assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
i:nat
H:(S i < S (Rlength r2))%nat
Hreci:(i < S (Rlength r2))%nat ->
pos_Rl (cons c (cons r1 r2)) i <=
pos_Rl (cons c (cons r1 r2)) (S i)
pos_Rl (cons c (cons r1 r2)) (S i) <=
pos_Rl (cons c (cons r1 r2)) (S (S i))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
pos_Rl (cons c (cons r1 r2)) 0 = Rmin c b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
pos_Rl (cons c (cons r1 r2))
  (Init.Nat.pred (Rlength (cons c (cons r1 r2)))) =
Rmax c b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
Rlength (cons c (cons r1 r2)) =
S (Rlength (cons r3 lf1))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons c (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  clear Hreci; apply (H2 (S i)); simpl; assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
pos_Rl (cons c (cons r1 r2)) 0 = Rmin c b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
pos_Rl (cons c (cons r1 r2))
  (Init.Nat.pred (Rlength (cons c (cons r1 r2)))) =
Rmax c b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
Rlength (cons c (cons r1 r2)) =
S (Rlength (cons r3 lf1))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons c (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  simpl; unfold Rmin; case (Rle_dec c b) as [|[]];
    [ reflexivity | elim H0; intros; assumption ].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
pos_Rl (cons c (cons r1 r2))
  (Init.Nat.pred (Rlength (cons c (cons r1 r2)))) =
Rmax c b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
Rlength (cons c (cons r1 r2)) =
S (Rlength (cons r3 lf1))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons c (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  replace (Rmax c b) with (Rmax a b).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
pos_Rl (cons c (cons r1 r2))
  (Init.Nat.pred (Rlength (cons c (cons r1 r2)))) =
Rmax a b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
Rmax a b = Rmax c b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
Rlength (cons c (cons r1 r2)) =
S (Rlength (cons r3 lf1))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons c (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  rewrite <- H3; reflexivity.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
Rmax a b = Rmax c b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
Rlength (cons c (cons r1 r2)) =
S (Rlength (cons r3 lf1))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons c (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  unfold Rmax; case (Rle_dec c b) as [|[]]; case (Rle_dec a b) as [|[]];
    [ reflexivity
      | elim H0; intros; apply Rle_trans with c; assumption
      | elim H0; intros; assumption
      | elim H0; intros; apply Rle_trans with c; assumption ].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
Rlength (cons c (cons r1 r2)) =
S (Rlength (cons r3 lf1))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons c (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  simpl; simpl in H5; apply H5.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
forall i : nat,
(i < Init.Nat.pred (Rlength (cons c (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  intros; simpl in H; induction  i as [| i Hreci].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H:(0 < S (Rlength r2))%nat
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) 0)
     (pos_Rl (cons c (cons r1 r2)) 1))
  (pos_Rl (cons r3 lf1) 0)
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
i:nat
H:(S i < S (Rlength r2))%nat
Hreci:(i < S (Rlength r2))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) (S i))
     (pos_Rl (cons c (cons r1 r2)) (S (S i))))
  (pos_Rl (cons r3 lf1) (S i))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  unfold constant_D_eq, open_interval; intros; simpl;
    apply (H7 0%nat).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H:(0 < S (Rlength r2))%nat
x:R
H6:pos_Rl (cons c (cons r1 r2)) 0 < x <
pos_Rl (cons c (cons r1 r2)) 1
(0 < Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H:(0 < S (Rlength r2))%nat
x:R
H6:pos_Rl (cons c (cons r1 r2)) 0 < x <
pos_Rl (cons c (cons r1 r2)) 1
open_interval (pos_Rl (cons r (cons r1 r2)) 0)
  (pos_Rl (cons r (cons r1 r2)) 1) x
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
i:nat
H:(S i < S (Rlength r2))%nat
Hreci:(i < S (Rlength r2))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) (S i))
     (pos_Rl (cons c (cons r1 r2)) (S (S i))))
  (pos_Rl (cons r3 lf1) (S i))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  simpl; apply lt_O_Sn.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H:(0 < S (Rlength r2))%nat
x:R
H6:pos_Rl (cons c (cons r1 r2)) 0 < x <
pos_Rl (cons c (cons r1 r2)) 1
open_interval (pos_Rl (cons r (cons r1 r2)) 0)
  (pos_Rl (cons r (cons r1 r2)) 1) x
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
i:nat
H:(S i < S (Rlength r2))%nat
Hreci:(i < S (Rlength r2))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) (S i))
     (pos_Rl (cons c (cons r1 r2)) (S (S i))))
  (pos_Rl (cons r3 lf1) (S i))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  unfold open_interval; simpl; simpl in H6; elim H6; clear H6;
    intros; split; try assumption; apply Rle_lt_trans with c;
      try assumption; replace r with a.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H:(0 < S (Rlength r2))%nat
x:R
H6:c < x
H8:x < r1
a <= c
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H:(0 < S (Rlength r2))%nat
x:R
H6:c < x
H8:x < r1
a = r
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
i:nat
H:(S i < S (Rlength r2))%nat
Hreci:(i < S (Rlength r2))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) (S i))
     (pos_Rl (cons c (cons r1 r2)) (S (S i))))
  (pos_Rl (cons r3 lf1) (S i))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  elim H0; intros; assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i : nat,
(i <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval (pos_Rl (cons r (cons r1 r2)) i)
     (pos_Rl (cons r (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
H:(0 < S (Rlength r2))%nat
x:R
H6:c < x
H8:x < r1
a = r
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
i:nat
H:(S i < S (Rlength r2))%nat
Hreci:(i < S (Rlength r2))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) (S i))
     (pos_Rl (cons c (cons r1 r2)) (S (S i))))
  (pos_Rl (cons r3 lf1) (S i))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  simpl in H4; rewrite H4; unfold Rmin; case (Rle_dec a b) as [|[]];
    [ reflexivity
      | elim H0; intros; apply Rle_trans with c; assumption ].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:c <= r1
H2:ordered_Rlist (cons r (cons r1 r2))
H4:pos_Rl (cons r (cons r1 r2)) 0 = Rmin a b
H3:pos_Rl (cons r (cons r1 r2))
  (Init.Nat.pred (Rlength (cons r (cons r1 r2)))) =
Rmax a b
H5:Rlength (cons r (cons r1 r2)) =
S (Rlength (cons r3 lf1))
H7:forall i0 : nat,
(i0 <
 Init.Nat.pred (Rlength (cons r (cons r1 r2))))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons r (cons r1 r2)) i0)
     (pos_Rl (cons r (cons r1 r2)) (S i0)))
  (pos_Rl (cons r3 lf1) i0)
i:nat
H:(S i < S (Rlength r2))%nat
Hreci:(i < S (Rlength r2))%nat ->
constant_D_eq f
  (open_interval
     (pos_Rl (cons c (cons r1 r2)) i)
     (pos_Rl (cons c (cons r1 r2)) (S i)))
  (pos_Rl (cons r3 lf1) i)
constant_D_eq f
  (open_interval (pos_Rl (cons c (cons r1 r2)) (S i))
     (pos_Rl (cons c (cons r1 r2)) (S (S i))))
  (pos_Rl (cons r3 lf1) (S i))
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  clear Hreci; apply (H7 (S i)); simpl; assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
  cut (adapted_couple f r1 b (cons r1 r2) lf1).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1 ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  cut (r1 <= c <= b).l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b ->
adapted_couple f r1 b (cons r1 r2) lf1 ->
{l : Rlist & {l0 : Rlist & adapted_couple f c b l l0}}
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  intros; elim (X0 _ _ _ _ _ H3 H2); intros l1' [lf1' H4]; split with l1';
    split with lf1'; assumption.l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
r1 <= c <= b
l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  split; [ left; assumption | elim H0; intros; assumption ].l1:Rlist
r:R
r0:Rlist
r1:R
r2:Rlist
X0:forall (lf0 : Rlist) (a1 b0 c0 : R) (f0 : R -> R),
adapted_couple f0 a1 b0 (cons r1 r2) lf0 ->
a1 <= c0 <= b0 ->
{l : Rlist &
{l0 : Rlist & adapted_couple f0 c0 b0 l l0}}
r3:R
lf1:Rlist
a, b, c:R
f:R -> R
H:adapted_couple f a b (cons r (cons r1 r2))
  (cons r3 lf1)
H0:a <= c <= b
H1:{c <= r1} + {r1 < c}
a0:r1 < c
adapted_couple f r1 b (cons r1 r2) lf1
  eapply StepFun_P7;
    [ elim H0; intros; apply Rle_trans with c; [ apply H2 | apply H3 ]
      | apply H ].
Qed.

Lemma StepFun_P46 :
  forall (f:R -> R) (a b c:R),
    IsStepFun f a b -> IsStepFun f b c -> IsStepFun f a c.forall (f : R -> R) (a b c : R),
IsStepFun f a b -> IsStepFun f b c -> IsStepFun f a c
Proof.forall (f : R -> R) (a b c : R),
IsStepFun f a b -> IsStepFun f b c -> IsStepFun f a c
  intros f; intros; case (Rle_dec a b); case (Rle_dec b c); intros.f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
r0:a <= b
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
r:a <= b
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  apply StepFun_P41 with b; assumption.f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
r:a <= b
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  case (Rle_dec a c); intro.f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
r:a <= b
r0:a <= c
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
r:a <= b
n0:~ a <= c
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  apply StepFun_P44 with b; try assumption.f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
r:a <= b
r0:a <= c
a <= c <= b
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
r:a <= b
n0:~ a <= c
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  split; [ assumption | auto with real ].f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
r:a <= b
n0:~ a <= c
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  apply StepFun_P6; apply StepFun_P44 with b.f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
r:a <= b
n0:~ a <= c
IsStepFun f c b
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
r:a <= b
n0:~ a <= c
c <= a <= b
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  apply StepFun_P6; assumption.f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
r:a <= b
n0:~ a <= c
c <= a <= b
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  split; auto with real.f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  case (Rle_dec a c); intro.f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
r0:a <= c
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
n0:~ a <= c
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  apply StepFun_P45 with b; try assumption.f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
r0:a <= c
b <= a <= c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
n0:~ a <= c
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  split; auto with real.f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
n0:~ a <= c
IsStepFun f a c
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  apply StepFun_P6; apply StepFun_P45 with b.f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
n0:~ a <= c
IsStepFun f b a
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
n0:~ a <= c
b <= c <= a
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  apply StepFun_P6; assumption.f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
r:b <= c
n:~ a <= b
n0:~ a <= c
b <= c <= a
f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  split; [ assumption | auto with real ].f:R -> R
a, b, c:R
X:IsStepFun f a b
X0:IsStepFun f b c
n:~ b <= c
n0:~ a <= b
IsStepFun f a c
  apply StepFun_P6; apply StepFun_P41 with b;
    auto with real || apply StepFun_P6; assumption.
Qed.