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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 Rseries.
Require Import SeqProp.
Require Import PartSum.
Require Import Max.
Local Open Scope R_scope.

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

Formalization of alternated series

Definition tg_alt (Un:nat -> R) (i:nat) : R := (-1) ^ i * Un i.
Definition positivity_seq (Un:nat -> R) : Prop := forall n:nat, 0 <= Un n.

Lemma CV_ALT_step0 :
  forall Un:nat -> R,
    Un_decreasing Un ->
    Un_growing (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))).forall Un : nat -> R,
Un_decreasing Un ->
Un_growing
  (fun N : nat => sum_f_R0 (tg_alt Un) (S (2 * N)))
Proof.forall Un : nat -> R,
Un_decreasing Un ->
Un_growing
  (fun N : nat => sum_f_R0 (tg_alt Un) (S (2 * N)))
  intros; unfold Un_growing; intro.Un:nat -> R
H:Un_decreasing Un
n:nat
sum_f_R0 (tg_alt Un) (S (2 * n)) <=
sum_f_R0 (tg_alt Un) (S (2 * S n))
  cut ((2 * S n)%nat = S (S (2 * n))).Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n)) ->
sum_f_R0 (tg_alt Un) (S (2 * n)) <=
sum_f_R0 (tg_alt Un) (S (2 * S n))
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  intro; rewrite H0.Un:nat -> R
H:Un_decreasing Un
n:nat
H0:(2 * S n)%nat = S (S (2 * n))
sum_f_R0 (tg_alt Un) (S (2 * n)) <=
sum_f_R0 (tg_alt Un) (S (S (S (2 * n))))
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  do 4 rewrite tech5; repeat rewrite Rplus_assoc; apply Rplus_le_compat_l.Un:nat -> R
H:Un_decreasing Un
n:nat
H0:(2 * S n)%nat = S (S (2 * n))
tg_alt Un (S (2 * n)) <=
tg_alt Un (S (2 * n)) +
(tg_alt Un (S (S (2 * n))) +
 tg_alt Un (S (S (S (2 * n)))))
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  pattern (tg_alt Un (S (2 * n))) at 1; rewrite <- Rplus_0_r.Un:nat -> R
H:Un_decreasing Un
n:nat
H0:(2 * S n)%nat = S (S (2 * n))
tg_alt Un (S (2 * n)) + 0 <=
tg_alt Un (S (2 * n)) +
(tg_alt Un (S (S (2 * n))) +
 tg_alt Un (S (S (S (2 * n)))))
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  apply Rplus_le_compat_l.Un:nat -> R
H:Un_decreasing Un
n:nat
H0:(2 * S n)%nat = S (S (2 * n))
0 <=
tg_alt Un (S (S (2 * n))) +
tg_alt Un (S (S (S (2 * n))))
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  unfold tg_alt; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even;
    rewrite Rmult_1_l.Un:nat -> R
H:Un_decreasing Un
n:nat
H0:(2 * S n)%nat = S (S (2 * n))
0 <= Un (2 * S n)%nat + -1 * Un (S (2 * S n))
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  apply Rplus_le_reg_l with (Un (S (2 * S n))).Un:nat -> R
H:Un_decreasing Un
n:nat
H0:(2 * S n)%nat = S (S (2 * n))
Un (S (2 * S n)) + 0 <=
Un (S (2 * S n)) +
(Un (2 * S n)%nat + -1 * Un (S (2 * S n)))
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  rewrite Rplus_0_r;
    replace (Un (S (2 * S n)) + (Un (2 * S n)%nat + -1 * Un (S (2 * S n)))) with
      (Un (2 * S n)%nat); [ idtac | ring ].Un:nat -> R
H:Un_decreasing Un
n:nat
H0:(2 * S n)%nat = S (S (2 * n))
Un (S (2 * S n)) <= Un (2 * S n)%nat
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  apply H.Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  cut (forall n:nat, S n = (n + 1)%nat); [ intro | intro; ring ].Un:nat -> R
H:Un_decreasing Un
n:nat
H0:forall n0 : nat, S n0 = (n0 + 1)%nat
(2 * S n)%nat = S (S (2 * n))
  rewrite (H0 n); rewrite (H0 (S (2 * n))); rewrite (H0 (2 * n)%nat); ring.
Qed.

Lemma CV_ALT_step1 :
  forall Un:nat -> R,
    Un_decreasing Un ->
    Un_decreasing (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N)).forall Un : nat -> R,
Un_decreasing Un ->
Un_decreasing
  (fun N : nat => sum_f_R0 (tg_alt Un) (2 * N))
Proof.forall Un : nat -> R,
Un_decreasing Un ->
Un_decreasing
  (fun N : nat => sum_f_R0 (tg_alt Un) (2 * N))
  intros; unfold Un_decreasing; intro.Un:nat -> R
H:Un_decreasing Un
n:nat
sum_f_R0 (tg_alt Un) (2 * S n) <=
sum_f_R0 (tg_alt Un) (2 * n)
  cut ((2 * S n)%nat = S (S (2 * n))).Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n)) ->
sum_f_R0 (tg_alt Un) (2 * S n) <=
sum_f_R0 (tg_alt Un) (2 * n)
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  intro; rewrite H0; do 2 rewrite tech5; repeat rewrite Rplus_assoc.Un:nat -> R
H:Un_decreasing Un
n:nat
H0:(2 * S n)%nat = S (S (2 * n))
sum_f_R0 (tg_alt Un) (2 * n) +
(tg_alt Un (S (2 * n)) + tg_alt Un (S (S (2 * n)))) <=
sum_f_R0 (tg_alt Un) (2 * n)
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  pattern (sum_f_R0 (tg_alt Un) (2 * n)) at 2; rewrite <- Rplus_0_r.Un:nat -> R
H:Un_decreasing Un
n:nat
H0:(2 * S n)%nat = S (S (2 * n))
sum_f_R0 (tg_alt Un) (2 * n) +
(tg_alt Un (S (2 * n)) + tg_alt Un (S (S (2 * n)))) <=
sum_f_R0 (tg_alt Un) (2 * n) + 0
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  apply Rplus_le_compat_l.Un:nat -> R
H:Un_decreasing Un
n:nat
H0:(2 * S n)%nat = S (S (2 * n))
tg_alt Un (S (2 * n)) + tg_alt Un (S (S (2 * n))) <= 0
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  unfold tg_alt; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even;
    rewrite Rmult_1_l.Un:nat -> R
H:Un_decreasing Un
n:nat
H0:(2 * S n)%nat = S (S (2 * n))
-1 * Un (S (2 * n)) + Un (2 * S n)%nat <= 0
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  apply Rplus_le_reg_l with (Un (S (2 * n))).Un:nat -> R
H:Un_decreasing Un
n:nat
H0:(2 * S n)%nat = S (S (2 * n))
Un (S (2 * n)) +
(-1 * Un (S (2 * n)) + Un (2 * S n)%nat) <=
Un (S (2 * n)) + 0
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  rewrite Rplus_0_r;
    replace (Un (S (2 * n)) + (-1 * Un (S (2 * n)) + Un (2 * S n)%nat)) with
      (Un (2 * S n)%nat); [ idtac | ring ].Un:nat -> R
H:Un_decreasing Un
n:nat
H0:(2 * S n)%nat = S (S (2 * n))
Un (2 * S n)%nat <= Un (S (2 * n))
Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  rewrite H0; apply H.Un:nat -> R
H:Un_decreasing Un
n:nat
(2 * S n)%nat = S (S (2 * n))
  cut (forall n:nat, S n = (n + 1)%nat); [ intro | intro; ring ].Un:nat -> R
H:Un_decreasing Un
n:nat
H0:forall n0 : nat, S n0 = (n0 + 1)%nat
(2 * S n)%nat = S (S (2 * n))
  rewrite (H0 n); rewrite (H0 (S (2 * n))); rewrite (H0 (2 * n)%nat); ring.
Qed.

(**********)
Lemma CV_ALT_step2 :
  forall (Un:nat -> R) (N:nat),
    Un_decreasing Un ->
    positivity_seq Un ->
    sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N)) <= 0.forall (Un : nat -> R) (N : nat),
Un_decreasing Un ->
positivity_seq Un ->
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) <=
0
Proof.forall (Un : nat -> R) (N : nat),
Un_decreasing Un ->
positivity_seq Un ->
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) <=
0
  intros; induction  N as [| N HrecN].Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * 0)) <=
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * S N)) <= 0
  simpl; unfold tg_alt; simpl; rewrite Rmult_1_r.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
-1 * Un 1%nat + -1 * -1 * Un 2%nat <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * S N)) <= 0
  replace (-1 * -1 * Un 2%nat) with (Un 2%nat); [ idtac | ring ].Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
-1 * Un 1%nat + Un 2%nat <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * S N)) <= 0
  apply Rplus_le_reg_l with (Un 1%nat); rewrite Rplus_0_r.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
Un 1%nat + (-1 * Un 1%nat + Un 2%nat) <= Un 1%nat
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * S N)) <= 0
  replace (Un 1%nat + (-1 * Un 1%nat + Un 2%nat)) with (Un 2%nat);
    [ apply H | ring ].Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * S N)) <= 0
  cut (S (2 * S N) = S (S (S (2 * N)))).Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N))) ->
sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * S N)) <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  intro; rewrite H1; do 2 rewrite tech5.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) +
tg_alt Un (S (S (S (2 * N)))) +
tg_alt Un (S (S (S (S (2 * N))))) <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  apply Rle_trans with (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N))).Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) +
tg_alt Un (S (S (S (2 * N)))) +
tg_alt Un (S (S (S (S (2 * N))))) <=
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N))
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) <=
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N))) at 2;
    rewrite <- Rplus_0_r.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) +
tg_alt Un (S (S (S (2 * N)))) +
tg_alt Un (S (S (S (S (2 * N))))) <=
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) +
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) <=
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  rewrite Rplus_assoc; apply Rplus_le_compat_l.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
tg_alt Un (S (S (S (2 * N)))) +
tg_alt Un (S (S (S (S (2 * N))))) <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) <=
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  unfold tg_alt; rewrite <- H1.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
(-1) ^ S (2 * S N) * Un (S (2 * S N)) +
(-1) ^ S (S (2 * S N)) * Un (S (S (2 * S N))) <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) <=
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  rewrite pow_1_odd.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
-1 * Un (S (2 * S N)) +
(-1) ^ S (S (2 * S N)) * Un (S (S (2 * S N))) <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) <=
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  cut (S (S (2 * S N)) = (2 * S (S N))%nat).Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
S (S (2 * S N)) = (2 * S (S N))%nat ->
-1 * Un (S (2 * S N)) +
(-1) ^ S (S (2 * S N)) * Un (S (S (2 * S N))) <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
S (S (2 * S N)) = (2 * S (S N))%nat
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) <=
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  intro; rewrite H2; rewrite pow_1_even; rewrite Rmult_1_l; rewrite <- H2.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
H2:S (S (2 * S N)) = (2 * S (S N))%nat
-1 * Un (S (2 * S N)) + Un (S (S (2 * S N))) <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
S (S (2 * S N)) = (2 * S (S N))%nat
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) <=
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  apply Rplus_le_reg_l with (Un (S (2 * S N))).Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
H2:S (S (2 * S N)) = (2 * S (S N))%nat
Un (S (2 * S N)) +
(-1 * Un (S (2 * S N)) + Un (S (S (2 * S N)))) <=
Un (S (2 * S N)) + 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
S (S (2 * S N)) = (2 * S (S N))%nat
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) <=
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  rewrite Rplus_0_r;
    replace (Un (S (2 * S N)) + (-1 * Un (S (2 * S N)) + Un (S (S (2 * S N)))))
      with (Un (S (S (2 * S N)))); [ idtac | ring ].Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
H2:S (S (2 * S N)) = (2 * S (S N))%nat
Un (S (S (2 * S N))) <= Un (S (2 * S N))
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
S (S (2 * S N)) = (2 * S (S N))%nat
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) <=
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  apply H.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
S (S (2 * S N)) = (2 * S (S N))%nat
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) <=
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  ring.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
H1:S (2 * S N) = S (S (S (2 * N)))
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * N)) <=
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  apply HrecN.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (S (2 * N)) <= 0
S (2 * S N) = S (S (S (2 * N)))
  ring.
Qed.

A more general inequality

Lemma CV_ALT_step3 :
  forall (Un:nat -> R) (N:nat),
    Un_decreasing Un ->
    positivity_seq Un -> sum_f_R0 (fun i:nat => tg_alt Un (S i)) N <= 0.forall (Un : nat -> R) (N : nat),
Un_decreasing Un ->
positivity_seq Un ->
sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <= 0
Proof.forall (Un : nat -> R) (N : nat),
Un_decreasing Un ->
positivity_seq Un ->
sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <= 0
  intros; induction  N as [| N HrecN].Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
sum_f_R0 (fun i : nat => tg_alt Un (S i)) 0 <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S N) <= 0
  simpl; unfold tg_alt; simpl; rewrite Rmult_1_r.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
-1 * Un 1%nat <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S N) <= 0
  apply Rplus_le_reg_l with (Un 1%nat).Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
Un 1%nat + -1 * Un 1%nat <= Un 1%nat + 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S N) <= 0
  rewrite Rplus_0_r; replace (Un 1%nat + -1 * Un 1%nat) with 0;
    [ apply H0 | ring ].Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S N) <= 0
  assert (H1 := even_odd_cor N).Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S N) <= 0
  elim H1; intros.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S N) <= 0
  elim H2; intro.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = (2 * x)%nat
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S N) <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S N) <= 0
  rewrite H3; apply CV_ALT_step2; assumption.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S N) <= 0
  rewrite H3; rewrite tech5.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) +
tg_alt Un (S (S (S (2 * x)))) <= 0
  apply Rle_trans with (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * x))).Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) +
tg_alt Un (S (S (S (2 * x)))) <=
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x))
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) <=
0
  pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * x))) at 2;
    rewrite <- Rplus_0_r.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) +
tg_alt Un (S (S (S (2 * x)))) <=
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) +
0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) <=
0
  apply Rplus_le_compat_l.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
tg_alt Un (S (S (S (2 * x)))) <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) <=
0
  unfold tg_alt; simpl.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
-1 * (-1 * (-1 * (-1) ^ (x + (x + 0)))) *
Un (S (S (S (x + (x + 0))))) <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) <=
0
  replace (x + (x + 0))%nat with (2 * x)%nat; [ idtac | ring ].Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
-1 * (-1 * (-1 * (-1) ^ (2 * x))) *
Un (S (S (S (2 * x)))) <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) <=
0
  rewrite pow_1_even.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
-1 * (-1 * (-1 * 1)) * Un (S (S (S (2 * x)))) <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) <=
0
  replace (-1 * (-1 * (-1 * 1)) * Un (S (S (S (2 * x))))) with
    (- Un (S (S (S (2 * x))))); [ idtac | ring ].Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
- Un (S (S (S (2 * x)))) <= 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) <=
0
  apply Rplus_le_reg_l with (Un (S (S (S (2 * x))))).Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
Un (S (S (S (2 * x)))) + - Un (S (S (S (2 * x)))) <=
Un (S (S (S (2 * x)))) + 0
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) <=
0
  rewrite Rplus_0_r; rewrite Rplus_opp_r.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
0 <= Un (S (S (S (2 * x))))
Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) <=
0
  apply H0.Un:nat -> R
N:nat
H:Un_decreasing Un
H0:positivity_seq Un
HrecN:sum_f_R0 (fun i : nat => tg_alt Un (S i)) N <=
0
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
x:nat
H2:N = (2 * x)%nat \/ N = S (2 * x)
H3:N = S (2 * x)
sum_f_R0 (fun i : nat => tg_alt Un (S i)) (S (2 * x)) <=
0
  apply CV_ALT_step2; assumption.
Qed.

  (**********)
Lemma CV_ALT_step4 :
  forall Un:nat -> R,
    Un_decreasing Un ->
    positivity_seq Un ->
    has_ub (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))).forall Un : nat -> R,
Un_decreasing Un ->
positivity_seq Un ->
has_ub
  (fun N : nat => sum_f_R0 (tg_alt Un) (S (2 * N)))
Proof.forall Un : nat -> R,
Un_decreasing Un ->
positivity_seq Un ->
has_ub
  (fun N : nat => sum_f_R0 (tg_alt Un) (S (2 * N)))
  intros; unfold has_ub; unfold bound.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
exists m : R,
  is_upper_bound
    (EUn
       (fun N : nat =>
        sum_f_R0 (tg_alt Un) (S (2 * N)))) m
  exists (Un 0%nat).Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
is_upper_bound
  (EUn
     (fun N : nat => sum_f_R0 (tg_alt Un) (S (2 * N))))
  (Un 0%nat)
  unfold is_upper_bound; intros; elim H1; intros.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
x <= Un 0%nat
  rewrite H2; rewrite decomp_sum.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
tg_alt Un 0 +
sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (Init.Nat.pred (S (2 * x0))) <= Un 0%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
(0 < S (2 * x0))%nat
  replace (tg_alt Un 0) with (Un 0%nat).Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
Un 0%nat +
sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (Init.Nat.pred (S (2 * x0))) <= Un 0%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
Un 0%nat = tg_alt Un 0
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
(0 < S (2 * x0))%nat
  pattern (Un 0%nat) at 2; rewrite <- Rplus_0_r.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
Un 0%nat +
sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (Init.Nat.pred (S (2 * x0))) <= Un 0%nat + 0
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
Un 0%nat = tg_alt Un 0
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
(0 < S (2 * x0))%nat
  apply Rplus_le_compat_l.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
sum_f_R0 (fun i : nat => tg_alt Un (S i))
  (Init.Nat.pred (S (2 * x0))) <= 0
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
Un 0%nat = tg_alt Un 0
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
(0 < S (2 * x0))%nat
  apply CV_ALT_step3; assumption.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
Un 0%nat = tg_alt Un 0
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
(0 < S (2 * x0))%nat
  unfold tg_alt; simpl; ring.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
x:R
H1:EUn
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N))) x
x0:nat
H2:x = sum_f_R0 (tg_alt Un) (S (2 * x0))
(0 < S (2 * x0))%nat
  apply lt_O_Sn.
Qed.

This lemma gives an interesting result about alternated series

Lemma CV_ALT :
  forall Un:nat -> R,
    Un_decreasing Un ->
    positivity_seq Un ->
    Un_cv Un 0 ->
    { l:R | Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l }.forall Un : nat -> R,
Un_decreasing Un ->
positivity_seq Un ->
Un_cv Un 0 ->
{l : R |
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N) l}
Proof.forall Un : nat -> R,
Un_decreasing Un ->
positivity_seq Un ->
Un_cv Un 0 ->
{l : R |
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N) l}
  intros.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:Un_cv Un 0
{l : R |
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N) l}
  assert (H2 := CV_ALT_step0 _ H).Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:Un_cv Un 0
H2:Un_growing
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
{l : R |
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N) l}
  assert (H3 := CV_ALT_step4 _ H H0).Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:Un_cv Un 0
H2:Un_growing
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
H3:has_ub
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
{l : R |
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N) l}
  destruct (growing_cv _ H2 H3) as (x,p).Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:Un_cv Un 0
H2:Un_growing
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
H3:has_ub
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
x:R
p:Un_cv
  (fun N : nat => sum_f_R0 (tg_alt Un) (S (2 * N)))
  x
{l : R |
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N) l}
  exists x.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:Un_cv Un 0
H2:Un_growing
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
H3:has_ub
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
x:R
p:Un_cv
  (fun N : nat => sum_f_R0 (tg_alt Un) (S (2 * N)))
  x
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N) x
  unfold Un_cv; unfold R_dist; unfold Un_cv in H1;
    unfold R_dist in H1; unfold Un_cv in p; unfold R_dist in p.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Un n - 0) < eps
H2:Un_growing
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
H3:has_ub
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
x:R
p:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n)) - x) <
  eps
forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
  intros; cut (0 < eps / 2);
    [ intro
      | unfold Rdiv; apply Rmult_lt_0_compat;
	[ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Un n - 0) < eps0
H2:Un_growing
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
H3:has_ub
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
  elim (H1 (eps / 2) H5); intros N2 H6.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Un n - 0) < eps0
H2:Un_growing
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
H3:has_ub
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n : nat,
(n >= N2)%nat -> Rabs (Un n - 0) < eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
  elim (p (eps / 2) H5); intros N1 H7.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Un n - 0) < eps0
H2:Un_growing
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
H3:has_ub
  (fun N : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n : nat,
(n >= N2)%nat -> Rabs (Un n - 0) < eps / 2
N1:nat
H7:forall n : nat,
(n >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n)) - x) <
eps / 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
  set (N := max (S (2 * N1)) N2).Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> Rabs (Un n - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n : nat,
(n >= N2)%nat -> Rabs (Un n - 0) < eps / 2
N1:nat
H7:forall n : nat,
(n >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
  exists N; intros.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
  assert (H9 := even_odd_cor n).Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
  elim H9; intros P H10.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
  cut (N1 <= P)%nat.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  intro; elim H10; intro.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  replace (sum_f_R0 (tg_alt Un) n - x) with
    (sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n)).Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
Rabs
  (sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n)) <
eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n) =
sum_f_R0 (tg_alt Un) n - x
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  apply Rle_lt_trans with
    (Rabs (sum_f_R0 (tg_alt Un) (S n) - x) + Rabs (- tg_alt Un (S n))).Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
Rabs
  (sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n)) <=
Rabs (sum_f_R0 (tg_alt Un) (S n) - x) +
Rabs (- tg_alt Un (S n))
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
Rabs (sum_f_R0 (tg_alt Un) (S n) - x) +
Rabs (- tg_alt Un (S n)) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n) =
sum_f_R0 (tg_alt Un) n - x
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  apply Rabs_triang.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
Rabs (sum_f_R0 (tg_alt Un) (S n) - x) +
Rabs (- tg_alt Un (S n)) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n) =
sum_f_R0 (tg_alt Un) n - x
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  rewrite (double_var eps); apply Rplus_lt_compat.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
Rabs (sum_f_R0 (tg_alt Un) (S n) - x) < eps / 2
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
Rabs (- tg_alt Un (S n)) < eps / 2
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n) =
sum_f_R0 (tg_alt Un) n - x
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  rewrite H12; apply H7; assumption.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
Rabs (- tg_alt Un (S n)) < eps / 2
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n) =
sum_f_R0 (tg_alt Un) n - x
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  rewrite Rabs_Ropp; unfold tg_alt; rewrite Rabs_mult;
    rewrite pow_1_abs; rewrite Rmult_1_l; unfold Rminus in H6;
      rewrite Ropp_0 in H6; rewrite <- (Rplus_0_r (Un (S n)));
	apply H6.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 + 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
(S n >= N2)%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n) =
sum_f_R0 (tg_alt Un) n - x
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  unfold ge; apply le_trans with n.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 + 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
(N2 <= n)%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 + 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
(n <= S n)%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n) =
sum_f_R0 (tg_alt Un) n - x
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  apply le_trans with N; [ unfold N; apply le_max_r | assumption ].Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 + 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
(n <= S n)%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n) =
sum_f_R0 (tg_alt Un) n - x
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  apply le_n_Sn.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = (2 * P)%nat
sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n) =
sum_f_R0 (tg_alt Un) n - x
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  rewrite tech5; ring.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
Rabs (sum_f_R0 (tg_alt Un) n - x) < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  rewrite H12; apply Rlt_trans with (eps / 2).Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
Rabs (sum_f_R0 (tg_alt Un) (S (2 * P)) - x) < eps / 2
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
eps / 2 < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  apply H7; assumption.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
eps / 2 < eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  unfold Rdiv; apply Rmult_lt_reg_l with 2.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
0 < 2
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
2 * (eps * / 2) < 2 * eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  prove_sup0.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
2 * (eps * / 2) < 2 * eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym;
    [ rewrite Rmult_1_r | discrR ].Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
eps < 2 * eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  rewrite double.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:(N1 <= P)%nat
H12:n = S (2 * P)
eps < eps + eps
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  pattern eps at 1; rewrite <- (Rplus_0_r eps); apply Rplus_lt_compat_l;
    assumption.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
(N1 <= P)%nat
  elim H10; intro; apply le_double.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = (2 * P)%nat
(2 * N1 <= 2 * P)%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(2 * N1 <= 2 * P)%nat
  rewrite <- H11; apply le_trans with N.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = (2 * P)%nat
(2 * N1 <= N)%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = (2 * P)%nat
(N <= n)%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(2 * N1 <= 2 * P)%nat
  unfold N; apply le_trans with (S (2 * N1));
    [ apply le_n_Sn | apply le_max_l ].Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = (2 * P)%nat
(N <= n)%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(2 * N1 <= 2 * P)%nat
  assumption.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(2 * N1 <= 2 * P)%nat
  apply lt_n_Sm_le.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(2 * N1 < S (2 * P))%nat
  rewrite <- H11.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(2 * N1 < n)%nat
  apply lt_le_trans with N.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(2 * N1 < N)%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(N <= n)%nat
  unfold N; apply lt_le_trans with (S (2 * N1)).Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(2 * N1 < S (2 * N1))%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(S (2 * N1) <= Nat.max (S (2 * N1)) N2)%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(N <= n)%nat
  apply lt_n_Sn.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(S (2 * N1) <= Nat.max (S (2 * N1)) N2)%nat
Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(N <= n)%nat
  apply le_max_l.Un:nat -> R
H:Un_decreasing Un
H0:positivity_seq Un
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - 0) < eps0
H2:Un_growing
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
H3:has_ub
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0)))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
  eps0
eps:R
H4:eps > 0
H5:0 < eps / 2
N2:nat
H6:forall n0 : nat,
(n0 >= N2)%nat -> Rabs (Un n0 - 0) < eps / 2
N1:nat
H7:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n0)) - x) <
eps / 2
N:=Nat.max (S (2 * N1)) N2:nat
n:nat
H8:(n >= N)%nat
H9:exists p0 : nat,
  n = (2 * p0)%nat \/ n = S (2 * p0)
P:nat
H10:n = (2 * P)%nat \/ n = S (2 * P)
H11:n = S (2 * P)
(N <= n)%nat
  assumption.
Qed.


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

Convergence of alternated series

Theorem alternated_series :
  forall Un:nat -> R,
    Un_decreasing Un ->
    Un_cv Un 0 ->
    { l:R | Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l }.forall Un : nat -> R,
Un_decreasing Un ->
Un_cv Un 0 ->
{l : R |
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N) l}
Proof.forall Un : nat -> R,
Un_decreasing Un ->
Un_cv Un 0 ->
{l : R |
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N) l}
  intros; apply CV_ALT.Un:nat -> R
H:Un_decreasing Un
H0:Un_cv Un 0
Un_decreasing Un
Un:nat -> R
H:Un_decreasing Un
H0:Un_cv Un 0
positivity_seq Un
Un:nat -> R
H:Un_decreasing Un
H0:Un_cv Un 0
Un_cv Un 0
  assumption.Un:nat -> R
H:Un_decreasing Un
H0:Un_cv Un 0
positivity_seq Un
Un:nat -> R
H:Un_decreasing Un
H0:Un_cv Un 0
Un_cv Un 0
  unfold positivity_seq; apply decreasing_ineq; assumption.Un:nat -> R
H:Un_decreasing Un
H0:Un_cv Un 0
Un_cv Un 0
  assumption.
Qed.

Theorem alternated_series_ineq :
  forall (Un:nat -> R) (l:R) (N:nat),
    Un_decreasing Un ->
    Un_cv Un 0 ->
    Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l ->
    sum_f_R0 (tg_alt Un) (S (2 * N)) <= l <= sum_f_R0 (tg_alt Un) (2 * N).forall (Un : nat -> R) (l : R) (N : nat),
Un_decreasing Un ->
Un_cv Un 0 ->
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l ->
sum_f_R0 (tg_alt Un) (S (2 * N)) <= l <=
sum_f_R0 (tg_alt Un) (2 * N)
Proof.forall (Un : nat -> R) (l : R) (N : nat),
Un_decreasing Un ->
Un_cv Un 0 ->
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l ->
sum_f_R0 (tg_alt Un) (S (2 * N)) <= l <=
sum_f_R0 (tg_alt Un) (2 * N)
  intros.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
sum_f_R0 (tg_alt Un) (S (2 * N)) <= l <=
sum_f_R0 (tg_alt Un) (2 * N)
  cut (Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N)) l).Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l ->
sum_f_R0 (tg_alt Un) (S (2 * N)) <= l <=
sum_f_R0 (tg_alt Un) (2 * N)
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  cut (Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))) l).Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (S (2 * N0)))
  l ->
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l ->
sum_f_R0 (tg_alt Un) (S (2 * N)) <= l <=
sum_f_R0 (tg_alt Un) (2 * N)
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (S (2 * N0)))
  l
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  intros; split.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
H2:Un_cv
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0))) l
H3:Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
sum_f_R0 (tg_alt Un) (S (2 * N)) <= l
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
H2:Un_cv
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0))) l
H3:Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
l <= sum_f_R0 (tg_alt Un) (2 * N)
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (S (2 * N0)))
  l
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  apply (growing_ineq (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N)))).Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
H2:Un_cv
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0))) l
H3:Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
Un_growing
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (S (2 * N0)))
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
H2:Un_cv
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0))) l
H3:Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (S (2 * N0)))
  l
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
H2:Un_cv
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0))) l
H3:Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
l <= sum_f_R0 (tg_alt Un) (2 * N)
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (S (2 * N0)))
  l
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  apply CV_ALT_step0; assumption.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
H2:Un_cv
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0))) l
H3:Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (S (2 * N0)))
  l
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
H2:Un_cv
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0))) l
H3:Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
l <= sum_f_R0 (tg_alt Un) (2 * N)
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (S (2 * N0)))
  l
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  assumption.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
H2:Un_cv
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0))) l
H3:Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
l <= sum_f_R0 (tg_alt Un) (2 * N)
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (S (2 * N0)))
  l
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  apply (decreasing_ineq (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N))).Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
H2:Un_cv
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0))) l
H3:Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
Un_decreasing
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
H2:Un_cv
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0))) l
H3:Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (S (2 * N0)))
  l
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  apply CV_ALT_step1; assumption.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
H2:Un_cv
  (fun N0 : nat =>
   sum_f_R0 (tg_alt Un) (S (2 * N0))) l
H3:Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (S (2 * N0)))
  l
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  assumption.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt Un) (S (2 * N0)))
  l
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  unfold Un_cv; unfold R_dist; unfold Un_cv in H1;
    unfold R_dist in H1; intros.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n - l) < eps0
eps:R
H2:eps > 0
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n)) - l) < eps
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  elim (H1 eps H2); intros.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n : nat,
(n >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n - l) < eps
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (S (2 * n)) - l) < eps
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  exists x; intros.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
Rabs (sum_f_R0 (tg_alt Un) (S (2 * n)) - l) < eps
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  apply H3.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(S (2 * n) >= x)%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  unfold ge; apply le_trans with (2 * n)%nat.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(x <= 2 * n)%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(2 * n <= S (2 * n))%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  apply le_trans with n.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(x <= n)%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(n <= 2 * n)%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(2 * n <= S (2 * n))%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  assumption.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(n <= 2 * n)%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(2 * n <= S (2 * n))%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  assert (H5 := mult_O_le n 2).Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
H5:2%nat = 0%nat \/ (n <= 2 * n)%nat
(n <= 2 * n)%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(2 * n <= S (2 * n))%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  elim H5; intro.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
H5:2%nat = 0%nat \/ (n <= 2 * n)%nat
H6:2%nat = 0%nat
(n <= 2 * n)%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
H5:2%nat = 0%nat \/ (n <= 2 * n)%nat
H6:(n <= 2 * n)%nat
(n <= 2 * n)%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(2 * n <= S (2 * n))%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  cut (0%nat <> 2%nat);
    [ intro; elim H7; symmetry ; assumption | discriminate ].Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
H5:2%nat = 0%nat \/ (n <= 2 * n)%nat
H6:(n <= 2 * n)%nat
(n <= 2 * n)%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(2 * n <= S (2 * n))%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  assumption.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(2 * n <= S (2 * n))%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  apply le_n_Sn.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) N0) l
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt Un) (2 * N0))
  l
  unfold Un_cv; unfold R_dist; unfold Un_cv in H1;
    unfold R_dist in H1; intros.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n - l) < eps0
eps:R
H2:eps > 0
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (2 * n) - l) < eps
  elim (H1 eps H2); intros.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n : nat,
(n >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n - l) < eps
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) (2 * n) - l) < eps
  exists x; intros.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
Rabs (sum_f_R0 (tg_alt Un) (2 * n) - l) < eps
  apply H3.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(2 * n >= x)%nat
  unfold ge; apply le_trans with n.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(x <= n)%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(n <= 2 * n)%nat
  assumption.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
(n <= 2 * n)%nat
  assert (H5 := mult_O_le n 2).Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
H5:2%nat = 0%nat \/ (n <= 2 * n)%nat
(n <= 2 * n)%nat
  elim H5; intro.Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
H5:2%nat = 0%nat \/ (n <= 2 * n)%nat
H6:2%nat = 0%nat
(n <= 2 * n)%nat
Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
H5:2%nat = 0%nat \/ (n <= 2 * n)%nat
H6:(n <= 2 * n)%nat
(n <= 2 * n)%nat
  cut (0%nat <> 2%nat);
    [ intro; elim H7; symmetry ; assumption | discriminate ].Un:nat -> R
l:R
N:nat
H:Un_decreasing Un
H0:Un_cv Un 0
H1:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps0
eps:R
H2:eps > 0
x:nat
H3:forall n0 : nat,
(n0 >= x)%nat ->
Rabs (sum_f_R0 (tg_alt Un) n0 - l) < eps
n:nat
H4:(n >= x)%nat
H5:2%nat = 0%nat \/ (n <= 2 * n)%nat
H6:(n <= 2 * n)%nat
(n <= 2 * n)%nat
  assumption.
Qed.

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

Application : construction of PI

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

Definition PI_tg (n:nat) := / INR (2 * n + 1).

Lemma PI_tg_pos : forall n:nat, 0 <= PI_tg n.forall n : nat, 0 <= PI_tg n
Proof.forall n : nat, 0 <= PI_tg n
  intro; unfold PI_tg; left; apply Rinv_0_lt_compat; apply lt_INR_0;
    replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ].
Qed.

Lemma PI_tg_decreasing : Un_decreasing PI_tg.Un_decreasing PI_tg
Proof.Un_decreasing PI_tg
  unfold PI_tg, Un_decreasing; intro.n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
  apply Rmult_le_reg_l with (INR (2 * n + 1)).n:nat
0 < INR (2 * n + 1)
n:nat
INR (2 * n + 1) * / INR (2 * S n + 1) <=
INR (2 * n + 1) * / INR (2 * n + 1)
  apply lt_INR_0.n:nat
(0 < 2 * n + 1)%nat
n:nat
INR (2 * n + 1) * / INR (2 * S n + 1) <=
INR (2 * n + 1) * / INR (2 * n + 1)
  replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ].n:nat
INR (2 * n + 1) * / INR (2 * S n + 1) <=
INR (2 * n + 1) * / INR (2 * n + 1)
  rewrite <- Rinv_r_sym.n:nat
INR (2 * n + 1) * / INR (2 * S n + 1) <= 1
n:nat
INR (2 * n + 1) <> 0
  apply Rmult_le_reg_l with (INR (2 * S n + 1)).n:nat
0 < INR (2 * S n + 1)
n:nat
INR (2 * S n + 1) *
(INR (2 * n + 1) * / INR (2 * S n + 1)) <=
INR (2 * S n + 1) * 1
n:nat
INR (2 * n + 1) <> 0
  apply lt_INR_0.n:nat
(0 < 2 * S n + 1)%nat
n:nat
INR (2 * S n + 1) *
(INR (2 * n + 1) * / INR (2 * S n + 1)) <=
INR (2 * S n + 1) * 1
n:nat
INR (2 * n + 1) <> 0
  replace (2 * S n + 1)%nat with (S (2 * S n)); [ apply lt_O_Sn | ring ].n:nat
INR (2 * S n + 1) *
(INR (2 * n + 1) * / INR (2 * S n + 1)) <=
INR (2 * S n + 1) * 1
n:nat
INR (2 * n + 1) <> 0
  rewrite (Rmult_comm (INR (2 * S n + 1))); rewrite Rmult_assoc;
    rewrite <- Rinv_l_sym.n:nat
INR (2 * n + 1) * 1 <= INR (2 * S n + 1) * 1
n:nat
INR (2 * S n + 1) <> 0
n:nat
INR (2 * n + 1) <> 0
  do 2 rewrite Rmult_1_r; apply le_INR.n:nat
(2 * n + 1 <= 2 * S n + 1)%nat
n:nat
INR (2 * S n + 1) <> 0
n:nat
INR (2 * n + 1) <> 0
  replace (2 * S n + 1)%nat with (S (S (2 * n + 1))).n:nat
(2 * n + 1 <= S (S (2 * n + 1)))%nat
n:nat
S (S (2 * n + 1)) = (2 * S n + 1)%nat
n:nat
INR (2 * S n + 1) <> 0
n:nat
INR (2 * n + 1) <> 0
  apply le_trans with (S (2 * n + 1)); apply le_n_Sn.n:nat
S (S (2 * n + 1)) = (2 * S n + 1)%nat
n:nat
INR (2 * S n + 1) <> 0
n:nat
INR (2 * n + 1) <> 0
  ring.n:nat
INR (2 * S n + 1) <> 0
n:nat
INR (2 * n + 1) <> 0
  apply not_O_INR; discriminate.n:nat
INR (2 * n + 1) <> 0
  apply not_O_INR; replace (2 * n + 1)%nat with (S (2 * n));
    [ discriminate | ring ].
Qed.

Lemma PI_tg_cv : Un_cv PI_tg 0.Un_cv PI_tg 0
Proof.Un_cv PI_tg 0
  unfold Un_cv; unfold R_dist; intros.eps:R
H:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (PI_tg n - 0) < eps
  cut (0 < 2 * eps);
    [ intro | apply Rmult_lt_0_compat; [ prove_sup0 | assumption ] ].eps:R
H:eps > 0
H0:0 < 2 * eps
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (PI_tg n - 0) < eps
  assert (H1 := archimed (/ (2 * eps))).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (PI_tg n - 0) < eps
  cut (0 <= up (/ (2 * eps)))%Z.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (PI_tg n - 0) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  intro; assert (H3 := IZN (up (/ (2 * eps))) H2).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (PI_tg n - 0) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  elim H3; intros N H4.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> Rabs (PI_tg n - 0) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  cut (0 < N)%nat.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> Rabs (PI_tg n - 0) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  intro; exists N; intros.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
Rabs (PI_tg n - 0) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  cut (0 < n)%nat.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat -> Rabs (PI_tg n - 0) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  intro; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r;
    rewrite Rabs_right.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  unfold PI_tg; apply Rlt_trans with (/ INR (2 * n)).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n + 1) < / INR (2 * n)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply Rmult_lt_reg_l with (INR (2 * n)).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
0 < INR (2 * n)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) * / INR (2 * n + 1) <
INR (2 * n) * / INR (2 * n)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply lt_INR_0.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
(0 < 2 * n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) * / INR (2 * n + 1) <
INR (2 * n) * / INR (2 * n)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  replace (2 * n)%nat with (n + n)%nat; [ idtac | ring ].eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
(0 < n + n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) * / INR (2 * n + 1) <
INR (2 * n) * / INR (2 * n)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply lt_le_trans with n.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
(n <= n + n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) * / INR (2 * n + 1) <
INR (2 * n) * / INR (2 * n)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  assumption.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
(n <= n + n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) * / INR (2 * n + 1) <
INR (2 * n) * / INR (2 * n)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply le_plus_l.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) * / INR (2 * n + 1) <
INR (2 * n) * / INR (2 * n)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  rewrite <- Rinv_r_sym.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) * / INR (2 * n + 1) < 1
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply Rmult_lt_reg_l with (INR (2 * n + 1)).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
0 < INR (2 * n + 1)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n + 1) * (INR (2 * n) * / INR (2 * n + 1)) <
INR (2 * n + 1) * 1
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply lt_INR_0.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
(0 < 2 * n + 1)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n + 1) * (INR (2 * n) * / INR (2 * n + 1)) <
INR (2 * n + 1) * 1
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ].eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n + 1) * (INR (2 * n) * / INR (2 * n + 1)) <
INR (2 * n + 1) * 1
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  rewrite (Rmult_comm (INR (2 * n + 1))).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) * / INR (2 * n + 1) * INR (2 * n + 1) <
INR (2 * n + 1) * 1
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  rewrite Rmult_assoc; rewrite <- Rinv_l_sym.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) * 1 < INR (2 * n + 1) * 1
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n + 1) <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  do 2 rewrite Rmult_1_r; apply lt_INR.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
(2 * n < 2 * n + 1)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n + 1) <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_n_Sn | ring ].eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n + 1) <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply not_O_INR; replace (2 * n + 1)%nat with (S (2 * n));
    [ discriminate | ring ].eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * n) <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  replace n with (S (pred n)).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * S (Init.Nat.pred n)) <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
S (Init.Nat.pred n) = n
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply not_O_INR; discriminate.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
S (Init.Nat.pred n) = n
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  symmetry ; apply S_pred with 0%nat.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  assumption.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply Rle_lt_trans with (/ INR (2 * N)).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * n) <= / INR (2 * N)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * N) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply Rinv_le_contravar.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
0 < INR (2 * N)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * N) <= INR (2 * n)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * N) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  rewrite mult_INR; apply Rmult_lt_0_compat;
    [ simpl; prove_sup0 | apply lt_INR_0; assumption ].eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR (2 * N) <= INR (2 * n)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * N) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply le_INR.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
(2 * N <= 2 * n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * N) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  now apply mult_le_compat_l.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ INR (2 * N) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  rewrite mult_INR.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ (INR 2 * INR N) < eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply Rmult_lt_reg_l with (INR N / eps).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
0 < INR N / eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR N / eps * / (INR 2 * INR N) < INR N / eps * eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply Rdiv_lt_0_compat with (2 := H).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
0 < INR N
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR N / eps * / (INR 2 * INR N) < INR N / eps * eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  now apply (lt_INR 0).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR N / eps * / (INR 2 * INR N) < INR N / eps * eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  replace (_ */ _) with (/(2 * eps)).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ (2 * eps) < INR N / eps * eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ (2 * eps) = INR N / eps * / (INR 2 * INR N)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  replace (_ / _ * _) with (INR N).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ (2 * eps) < INR N
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR N = INR N / eps * eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ (2 * eps) = INR N / eps * / (INR 2 * INR N)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  rewrite INR_IZR_INZ.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ (2 * eps) < IZR (Z.of_nat N)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR N = INR N / eps * eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ (2 * eps) = INR N / eps * / (INR 2 * INR N)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  now rewrite <- H4.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR N = INR N / eps * eps
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ (2 * eps) = INR N / eps * / (INR 2 * INR N)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  field.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
eps <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ (2 * eps) = INR N / eps * / (INR 2 * INR N)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  now apply Rgt_not_eq.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
/ (2 * eps) = INR N / eps * / (INR 2 * INR N)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  simpl (INR 2); field; split.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
INR N <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
eps <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  now apply Rgt_not_eq, (lt_INR 0).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
eps <> 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  now apply Rgt_not_eq.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
H7:(0 < n)%nat
PI_tg n >= 0
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply Rle_ge; apply PI_tg_pos.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:(0 < N)%nat
n:nat
H6:(n >= N)%nat
(0 < n)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply lt_le_trans with N; assumption.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  elim H1; intros H5 _.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:IZR (up (/ (2 * eps))) > / (2 * eps)
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  destruct (lt_eq_lt_dec 0 N) as [[| <- ]|H6].eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:IZR (up (/ (2 * eps))) > / (2 * eps)
l:(0 < N)%nat
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
H4:up (/ (2 * eps)) = Z.of_nat 0
H5:IZR (up (/ (2 * eps))) > / (2 * eps)
(0 < 0)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:IZR (up (/ (2 * eps))) > / (2 * eps)
H6:(N < 0)%nat
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  assumption.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
H4:up (/ (2 * eps)) = Z.of_nat 0
H5:IZR (up (/ (2 * eps))) > / (2 * eps)
(0 < 0)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:IZR (up (/ (2 * eps))) > / (2 * eps)
H6:(N < 0)%nat
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  rewrite H4 in H5.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
H4:up (/ (2 * eps)) = Z.of_nat 0
H5:IZR (Z.of_nat 0) > / (2 * eps)
(0 < 0)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:IZR (up (/ (2 * eps))) > / (2 * eps)
H6:(N < 0)%nat
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  simpl in H5.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
H4:up (/ (2 * eps)) = Z.of_nat 0
H5:0 > / (2 * eps)
(0 < 0)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:IZR (up (/ (2 * eps))) > / (2 * eps)
H6:(N < 0)%nat
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  cut (0 < / (2 * eps)); [ intro | apply Rinv_0_lt_compat; assumption ].eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
H4:up (/ (2 * eps)) = Z.of_nat 0
H5:0 > / (2 * eps)
H6:0 < / (2 * eps)
(0 < 0)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:IZR (up (/ (2 * eps))) > / (2 * eps)
H6:(N < 0)%nat
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  elim (Rlt_irrefl _ (Rlt_trans _ _ _ H6 H5)).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
H2:(0 <= up (/ (2 * eps)))%Z
H3:exists m : nat, up (/ (2 * eps)) = Z.of_nat m
N:nat
H4:up (/ (2 * eps)) = Z.of_nat N
H5:IZR (up (/ (2 * eps))) > / (2 * eps)
H6:(N < 0)%nat
(0 < N)%nat
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  elim (lt_n_O _ H6).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
(0 <= up (/ (2 * eps)))%Z
  apply le_IZR.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
0 <= IZR (up (/ (2 * eps)))
  left; apply Rlt_trans with (/ (2 * eps)).eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
0 < / (2 * eps)
eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
/ (2 * eps) < IZR (up (/ (2 * eps)))
  apply Rinv_0_lt_compat; assumption.eps:R
H:eps > 0
H0:0 < 2 * eps
H1:IZR (up (/ (2 * eps))) > / (2 * eps) /\
IZR (up (/ (2 * eps))) - / (2 * eps) <= 1
/ (2 * eps) < IZR (up (/ (2 * eps)))
  elim H1; intros; assumption.
Qed.

Lemma exist_PI :
  { l:R | Un_cv (fun N:nat => sum_f_R0 (tg_alt PI_tg) N) l }.{l : R |
Un_cv (fun N : nat => sum_f_R0 (tg_alt PI_tg) N) l}
Proof.{l : R |
Un_cv (fun N : nat => sum_f_R0 (tg_alt PI_tg) N) l}
  apply alternated_series.Un_decreasing PI_tg
Un_cv PI_tg 0
  apply PI_tg_decreasing.Un_cv PI_tg 0
  apply PI_tg_cv.
Qed.

Now, PI is defined

Definition Alt_PI : R := 4 * (let (a,_) := exist_PI in a).

We can get an approximation of PI with the following inequality

Lemma Alt_PI_ineq :
  forall N:nat,
    sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= Alt_PI / 4 <=
    sum_f_R0 (tg_alt PI_tg) (2 * N).forall N : nat,
sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= Alt_PI / 4 <=
sum_f_R0 (tg_alt PI_tg) (2 * N)
Proof.forall N : nat,
sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= Alt_PI / 4 <=
sum_f_R0 (tg_alt PI_tg) (2 * N)
  intro; apply alternated_series_ineq.N:nat
Un_decreasing PI_tg
N:nat
Un_cv PI_tg 0
N:nat
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt PI_tg) N0)
  (Alt_PI / 4)
  apply PI_tg_decreasing.N:nat
Un_cv PI_tg 0
N:nat
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt PI_tg) N0)
  (Alt_PI / 4)
  apply PI_tg_cv.N:nat
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt PI_tg) N0)
  (Alt_PI / 4)
  unfold Alt_PI; case exist_PI; intro.N:nat
x:R
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt PI_tg) N0) x ->
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt PI_tg) N0)
  (4 * x / 4)
  replace (4 * x / 4) with x.N:nat
x:R
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt PI_tg) N0) x ->
Un_cv (fun N0 : nat => sum_f_R0 (tg_alt PI_tg) N0) x
N:nat
x:R
x = 4 * x / 4
  trivial.N:nat
x:R
x = 4 * x / 4
  unfold Rdiv; rewrite (Rmult_comm 4); rewrite Rmult_assoc;
    rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r; reflexivity | discrR ].
Qed.

Lemma Alt_PI_RGT_0 : 0 < Alt_PI.0 < Alt_PI
Proof.0 < Alt_PI
  assert (H := Alt_PI_ineq 0).H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI / 4 <= sum_f_R0 (tg_alt PI_tg) (2 * 0)
0 < Alt_PI
  apply Rmult_lt_reg_l with (/ 4).H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI / 4 <= sum_f_R0 (tg_alt PI_tg) (2 * 0)
0 < / 4
H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI / 4 <= sum_f_R0 (tg_alt PI_tg) (2 * 0)
/ 4 * 0 < / 4 * Alt_PI
  apply Rinv_0_lt_compat; prove_sup0.H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI / 4 <= sum_f_R0 (tg_alt PI_tg) (2 * 0)
/ 4 * 0 < / 4 * Alt_PI
  rewrite Rmult_0_r; rewrite Rmult_comm.H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI / 4 <= sum_f_R0 (tg_alt PI_tg) (2 * 0)
0 < Alt_PI * / 4
  elim H; clear H; intros H _.H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <= Alt_PI / 4
0 < Alt_PI * / 4
  unfold Rdiv in H;
    apply Rlt_le_trans with (sum_f_R0 (tg_alt PI_tg) (S (2 * 0))).H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI * / 4
0 < sum_f_R0 (tg_alt PI_tg) (S (2 * 0))
H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI * / 4
sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <= Alt_PI * / 4
  simpl; unfold tg_alt; simpl; rewrite Rmult_1_l;
    rewrite Rmult_1_r; apply Rplus_lt_reg_l with (PI_tg 1).H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI * / 4
PI_tg 1 + 0 < PI_tg 1 + (PI_tg 0 + -1 * PI_tg 1)
H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI * / 4
sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <= Alt_PI * / 4
  rewrite Rplus_0_r;
    replace (PI_tg 1 + (PI_tg 0 + -1 * PI_tg 1)) with (PI_tg 0);
      [ unfold PI_tg | ring ].H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI * / 4
/ INR (2 * 1 + 1) < / INR (2 * 0 + 1)
H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI * / 4
sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <= Alt_PI * / 4
  simpl; apply Rinv_lt_contravar.H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI * / 4
0 < 1 * (1 + 1 + 1)
H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI * / 4
1 < 1 + 1 + 1
H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI * / 4
sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <= Alt_PI * / 4
  rewrite Rmult_1_l; replace (2 + 1) with 3; [ prove_sup0 | ring ].H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI * / 4
1 < 1 + 1 + 1
H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI * / 4
sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <= Alt_PI * / 4
  rewrite Rplus_comm; pattern 1 at 1; rewrite <- Rplus_0_r;
    apply Rplus_lt_compat_l; prove_sup0.H:sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <=
Alt_PI * / 4
sum_f_R0 (tg_alt PI_tg) (S (2 * 0)) <= Alt_PI * / 4
  assumption.
Qed.