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

(*i Due to L.Thery i*)

(************************************************************)
(* Definitions of log and Rpower : R->R->R; main properties *)
(************************************************************)

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo1.
Require Import Ranalysis1.
Require Import Exp_prop.
Require Import Rsqrt_def.
Require Import R_sqrt.
Require Import Sqrt_reg.
Require Import MVT.
Require Import Ranalysis4.
Require Import Lra.
Local Open Scope R_scope.

Lemma P_Rmin : forall (P:R -> Prop) (x y:R), P x -> P y -> P (Rmin x y).forall (P : R -> Prop) (x y : R),
P x -> P y -> P (Rmin x y)
Proof.forall (P : R -> Prop) (x y : R),
P x -> P y -> P (Rmin x y)
  intros P x y H1 H2; unfold Rmin; case (Rle_dec x y); intro;
    assumption.
Qed.

Lemma exp_le_3 : exp 1 <= 3.exp 1 <= 3
Proof.exp 1 <= 3
  assert (exp_1 : exp 1 <> 0).exp 1 <> 0
exp_1:exp 1 <> 0
exp 1 <= 3
  assert (H0 := exp_pos 1); red; intro; rewrite H in H0;
    elim (Rlt_irrefl _ H0).exp_1:exp 1 <> 0
exp 1 <= 3
  apply Rmult_le_reg_l with (/ exp 1).exp_1:exp 1 <> 0
0 < / exp 1
exp_1:exp 1 <> 0
/ exp 1 * exp 1 <= / exp 1 * 3
  apply Rinv_0_lt_compat; apply exp_pos.exp_1:exp 1 <> 0
/ exp 1 * exp 1 <= / exp 1 * 3
  rewrite <- Rinv_l_sym.exp_1:exp 1 <> 0
1 <= / exp 1 * 3
exp_1:exp 1 <> 0
exp 1 <> 0
  apply Rmult_le_reg_l with (/ 3).exp_1:exp 1 <> 0
0 < / 3
exp_1:exp 1 <> 0
/ 3 * 1 <= / 3 * (/ exp 1 * 3)
exp_1:exp 1 <> 0
exp 1 <> 0
  apply Rinv_0_lt_compat; prove_sup0.exp_1:exp 1 <> 0
/ 3 * 1 <= / 3 * (/ exp 1 * 3)
exp_1:exp 1 <> 0
exp 1 <> 0
  rewrite Rmult_1_r; rewrite <- (Rmult_comm 3); rewrite <- Rmult_assoc;
    rewrite <- Rinv_l_sym.exp_1:exp 1 <> 0
/ 3 <= 1 * / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  rewrite Rmult_1_l; replace (/ exp 1) with (exp (-1)).exp_1:exp 1 <> 0
/ 3 <= exp (-1)
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  unfold exp; case (exist_exp (-1)) as (?,e); simpl in |- *;
    unfold exp_in in e;
      assert (H := alternated_series_ineq (fun i:nat => / INR (fact i)) x 1).exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
/ 3 <= x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  cut
    (sum_f_R0 (tg_alt (fun i:nat => / INR (fact i))) (S (2 * 1)) <= x <=
      sum_f_R0 (tg_alt (fun i:nat => / INR (fact i))) (2 * 1)).exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1) -> / 3 <= x
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  intro; elim H0; clear H0; intros H0 _; simpl in H0; unfold tg_alt in H0;
    simpl in H0.exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
H0:1 * / 1 + -1 * 1 * / 1 +
-1 * (-1 * 1) * / (1 + 1) +
-1 * (-1 * (-1 * 1)) * / (1 + 1 + 1 + 1 + 1 + 1) <=
x
/ 3 <= x
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  replace (/ 3) with
  (1 * / 1 + -1 * 1 * / 1 + -1 * (-1 * 1) * / 2 +
    -1 * (-1 * (-1 * 1)) * / (2 + 1 + 1 + 1 + 1)) by field.exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
H0:1 * / 1 + -1 * 1 * / 1 +
-1 * (-1 * 1) * / (1 + 1) +
-1 * (-1 * (-1 * 1)) * / (1 + 1 + 1 + 1 + 1 + 1) <=
x
1 * / 1 + -1 * 1 * / 1 + -1 * (-1 * 1) * / 2 +
-1 * (-1 * (-1 * 1)) * / (2 + 1 + 1 + 1 + 1) <= x
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  apply H0.exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  apply H.exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_decreasing (fun i : nat => / INR (fact i))
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv (fun i : nat => / INR (fact i)) 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (fun i : nat => / INR (fact i))) N)
  x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  unfold Un_decreasing; intros;
    apply Rmult_le_reg_l with (INR (fact n)).exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
0 < INR (fact n)
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
INR (fact n) * / INR (fact (S n)) <=
INR (fact n) * / INR (fact n)
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv (fun i : nat => / INR (fact i)) 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (fun i : nat => / INR (fact i))) N)
  x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  apply INR_fact_lt_0.exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
INR (fact n) * / INR (fact (S n)) <=
INR (fact n) * / INR (fact n)
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv (fun i : nat => / INR (fact i)) 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (fun i : nat => / INR (fact i))) N)
  x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  apply Rmult_le_reg_l with (INR (fact (S n))).exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
0 < INR (fact (S n))
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
INR (fact (S n)) * (INR (fact n) * / INR (fact (S n))) <=
INR (fact (S n)) * (INR (fact n) * / INR (fact n))
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv (fun i : nat => / INR (fact i)) 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (fun i : nat => / INR (fact i))) N)
  x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  apply INR_fact_lt_0.exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
INR (fact (S n)) * (INR (fact n) * / INR (fact (S n))) <=
INR (fact (S n)) * (INR (fact n) * / INR (fact n))
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv (fun i : nat => / INR (fact i)) 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (fun i : nat => / INR (fact i))) N)
  x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  rewrite <- Rinv_r_sym.exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
INR (fact (S n)) * (INR (fact n) * / INR (fact (S n))) <=
INR (fact (S n)) * 1
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
INR (fact n) <> 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv (fun i : nat => / INR (fact i)) 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (fun i : nat => / INR (fact i))) N)
  x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc;
    rewrite <- Rinv_l_sym.exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
INR (fact n) * 1 <= INR (fact (S n))
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
INR (fact (S n)) <> 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
INR (fact n) <> 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv (fun i : nat => / INR (fact i)) 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (fun i : nat => / INR (fact i))) N)
  x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  rewrite Rmult_1_r; apply le_INR; apply fact_le; apply le_n_Sn.exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
INR (fact (S n)) <> 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
INR (fact n) <> 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv (fun i : nat => / INR (fact i)) 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (fun i : nat => / INR (fact i))) N)
  x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  apply INR_fact_neq_0.exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
n:nat
INR (fact n) <> 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv (fun i : nat => / INR (fact i)) 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (fun i : nat => / INR (fact i))) N)
  x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  apply INR_fact_neq_0.exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv (fun i : nat => / INR (fact i)) 0
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (fun i : nat => / INR (fact i))) N)
  x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  assert (H0 := cv_speed_pow_fact 1); unfold Un_cv; unfold Un_cv in H0;
    intros; elim (H0 _ H1); intros; exists x0; intros;
      unfold R_dist in H2; unfold R_dist;
        replace (/ INR (fact n)) with (1 ^ n / INR (fact n)).exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (1 ^ n0 / INR (fact n0)) 0 < eps0
eps:R
H1:eps > 0
x0:nat
H2:forall n0 : nat,
(n0 >= x0)%nat ->
Rabs (1 ^ n0 / INR (fact n0) - 0) < eps
n:nat
H3:(n >= x0)%nat
Rabs (1 ^ n / INR (fact n) - 0) < eps
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (1 ^ n0 / INR (fact n0)) 0 < eps0
eps:R
H1:eps > 0
x0:nat
H2:forall n0 : nat,
(n0 >= x0)%nat ->
Rabs (1 ^ n0 / INR (fact n0) - 0) < eps
n:nat
H3:(n >= x0)%nat
1 ^ n / INR (fact n) = / INR (fact n)
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (fun i : nat => / INR (fact i))) N)
  x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  apply (H2 _ H3).exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (1 ^ n0 / INR (fact n0)) 0 < eps0
eps:R
H1:eps > 0
x0:nat
H2:forall n0 : nat,
(n0 >= x0)%nat ->
Rabs (1 ^ n0 / INR (fact n0) - 0) < eps
n:nat
H3:(n >= x0)%nat
1 ^ n / INR (fact n) = / INR (fact n)
exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (fun i : nat => / INR (fact i))) N)
  x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  unfold Rdiv; rewrite pow1; rewrite Rmult_1_l; reflexivity.exp_1:exp 1 <> 0
x:R
e:infinite_sum
  (fun i : nat => / INR (fact i) * (-1) ^ i) x
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (fun i : nat => / INR (fact i))) N)
  x
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  unfold infinite_sum in e; unfold Un_cv, tg_alt; intros; elim (e _ H0);
    intros; exists x0; intros;
      replace (sum_f_R0 (fun i:nat => (-1) ^ i * / INR (fact i)) n) with
      (sum_f_R0 (fun i:nat => / INR (fact i) * (-1) ^ i) n).exp_1:exp 1 <> 0
x:R
e:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * (-1) ^ i)
       n0) x < eps0
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * (-1) ^ i)
     n0) x < eps
n:nat
H2:(n >= x0)%nat
R_dist
  (sum_f_R0 (fun i : nat => / INR (fact i) * (-1) ^ i)
     n) x < eps
exp_1:exp 1 <> 0
x:R
e:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * (-1) ^ i)
       n0) x < eps0
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * (-1) ^ i)
     n0) x < eps
n:nat
H2:(n >= x0)%nat
sum_f_R0 (fun i : nat => / INR (fact i) * (-1) ^ i) n =
sum_f_R0 (fun i : nat => (-1) ^ i * / INR (fact i)) n
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  apply (H1 _ H2).exp_1:exp 1 <> 0
x:R
e:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * (-1) ^ i)
       n0) x < eps0
H:Un_decreasing (fun i : nat => / INR (fact i)) ->
Un_cv (fun i : nat => / INR (fact i)) 0 ->
Un_cv
  (fun N : nat =>
   sum_f_R0
     (tg_alt (fun i : nat => / INR (fact i))) N) x ->
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (S (2 * 1)) <= x <=
sum_f_R0 (tg_alt (fun i : nat => / INR (fact i)))
  (2 * 1)
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * (-1) ^ i)
     n0) x < eps
n:nat
H2:(n >= x0)%nat
sum_f_R0 (fun i : nat => / INR (fact i) * (-1) ^ i) n =
sum_f_R0 (fun i : nat => (-1) ^ i * / INR (fact i)) n
exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  apply sum_eq; intros; apply Rmult_comm.exp_1:exp 1 <> 0
exp (-1) = / exp 1
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  apply Rmult_eq_reg_l with (exp 1).exp_1:exp 1 <> 0
exp 1 * exp (-1) = exp 1 * / exp 1
exp_1:exp 1 <> 0
exp 1 <> 0
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0;
    rewrite <- Rinv_r_sym.exp_1:exp 1 <> 0
1 = 1
exp_1:exp 1 <> 0
exp 1 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  reflexivity.exp_1:exp 1 <> 0
exp 1 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  assumption.exp_1:exp 1 <> 0
exp 1 <> 0
exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  assumption.exp_1:exp 1 <> 0
3 <> 0
exp_1:exp 1 <> 0
exp 1 <> 0
  discrR.exp_1:exp 1 <> 0
exp 1 <> 0
  assumption.
Qed.

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

Properties of Exp

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

Theorem exp_increasing : forall x y:R, x < y -> exp x < exp y.forall x y : R, x < y -> exp x < exp y
Proof.forall x y : R, x < y -> exp x < exp y
  intros x y H.x, y:R
H:x < y
exp x < exp y
  assert (H0 : derivable exp).x, y:R
H:x < y
derivable exp
x, y:R
H:x < y
H0:derivable exp
exp x < exp y
  apply derivable_exp.x, y:R
H:x < y
H0:derivable exp
exp x < exp y
  assert (H1 := positive_derivative _ H0).x, y:R
H:x < y
H0:derivable exp
H1:(forall x0 : R, 0 < derive_pt exp x0 (H0 x0)) ->
strict_increasing exp
exp x < exp y
  unfold strict_increasing in H1.x, y:R
H:x < y
H0:derivable exp
H1:(forall x0 : R, 0 < derive_pt exp x0 (H0 x0)) ->
forall x0 y0 : R, x0 < y0 -> exp x0 < exp y0
exp x < exp y
  apply H1.x, y:R
H:x < y
H0:derivable exp
H1:(forall x0 : R, 0 < derive_pt exp x0 (H0 x0)) ->
forall x0 y0 : R, x0 < y0 -> exp x0 < exp y0
forall x0 : R, 0 < derive_pt exp x0 (H0 x0)
x, y:R
H:x < y
H0:derivable exp
H1:(forall x0 : R, 0 < derive_pt exp x0 (H0 x0)) ->
forall x0 y0 : R, x0 < y0 -> exp x0 < exp y0
x < y
  intro.x, y:R
H:x < y
H0:derivable exp
H1:(forall x1 : R, 0 < derive_pt exp x1 (H0 x1)) ->
forall x1 y0 : R, x1 < y0 -> exp x1 < exp y0
x0:R
0 < derive_pt exp x0 (H0 x0)
x, y:R
H:x < y
H0:derivable exp
H1:(forall x0 : R, 0 < derive_pt exp x0 (H0 x0)) ->
forall x0 y0 : R, x0 < y0 -> exp x0 < exp y0
x < y
  replace (derive_pt exp x0 (H0 x0)) with (exp x0).x, y:R
H:x < y
H0:derivable exp
H1:(forall x1 : R, 0 < derive_pt exp x1 (H0 x1)) ->
forall x1 y0 : R, x1 < y0 -> exp x1 < exp y0
x0:R
0 < exp x0
x, y:R
H:x < y
H0:derivable exp
H1:(forall x1 : R, 0 < derive_pt exp x1 (H0 x1)) ->
forall x1 y0 : R, x1 < y0 -> exp x1 < exp y0
x0:R
exp x0 = derive_pt exp x0 (H0 x0)
x, y:R
H:x < y
H0:derivable exp
H1:(forall x0 : R, 0 < derive_pt exp x0 (H0 x0)) ->
forall x0 y0 : R, x0 < y0 -> exp x0 < exp y0
x < y
  apply exp_pos.x, y:R
H:x < y
H0:derivable exp
H1:(forall x1 : R, 0 < derive_pt exp x1 (H0 x1)) ->
forall x1 y0 : R, x1 < y0 -> exp x1 < exp y0
x0:R
exp x0 = derive_pt exp x0 (H0 x0)
x, y:R
H:x < y
H0:derivable exp
H1:(forall x0 : R, 0 < derive_pt exp x0 (H0 x0)) ->
forall x0 y0 : R, x0 < y0 -> exp x0 < exp y0
x < y
  symmetry ; apply derive_pt_eq_0.x, y:R
H:x < y
H0:derivable exp
H1:(forall x1 : R, 0 < derive_pt exp x1 (H0 x1)) ->
forall x1 y0 : R, x1 < y0 -> exp x1 < exp y0
x0:R
derivable_pt_lim exp x0 (exp x0)
x, y:R
H:x < y
H0:derivable exp
H1:(forall x0 : R, 0 < derive_pt exp x0 (H0 x0)) ->
forall x0 y0 : R, x0 < y0 -> exp x0 < exp y0
x < y
  apply (derivable_pt_lim_exp x0).x, y:R
H:x < y
H0:derivable exp
H1:(forall x0 : R, 0 < derive_pt exp x0 (H0 x0)) ->
forall x0 y0 : R, x0 < y0 -> exp x0 < exp y0
x < y
  apply H.
Qed.

Theorem exp_lt_inv : forall x y:R, exp x < exp y -> x < y.forall x y : R, exp x < exp y -> x < y
Proof.forall x y : R, exp x < exp y -> x < y
  intros x y H; case (Rtotal_order x y); [ intros H1 | intros [H1| H1] ].x, y:R
H:exp x < exp y
H1:x < y
x < y
x, y:R
H:exp x < exp y
H1:x = y
x < y
x, y:R
H:exp x < exp y
H1:x > y
x < y
  assumption.x, y:R
H:exp x < exp y
H1:x = y
x < y
x, y:R
H:exp x < exp y
H1:x > y
x < y
  rewrite H1 in H; elim (Rlt_irrefl _ H).x, y:R
H:exp x < exp y
H1:x > y
x < y
  assert (H2 := exp_increasing _ _ H1).x, y:R
H:exp x < exp y
H1:x > y
H2:exp y < exp x
x < y
  elim (Rlt_irrefl _ (Rlt_trans _ _ _ H H2)).
Qed.

Lemma exp_ineq1 : forall x:R, 0 < x -> 1 + x < exp x.forall x : R, 0 < x -> 1 + x < exp x
Proof.forall x : R, 0 < x -> 1 + x < exp x
  intros; apply Rplus_lt_reg_l with (- exp 0); rewrite <- (Rplus_comm (exp x));
    assert (H0 := MVT_cor1 exp 0 x derivable_exp H); elim H0;
      intros; elim H1; intros; unfold Rminus in H2; rewrite H2;
        rewrite Ropp_0; rewrite Rplus_0_r;
          replace (derive_pt exp x0 (derivable_exp x0)) with (exp x0).x:R
H:0 < x
H0:exists c : R,
  exp x - exp 0 =
  derive_pt exp c (derivable_exp c) * (x - 0) /\
  0 < c < x
x0:R
H1:exp x - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x - 0) /\
0 < x0 < x
H2:exp x + - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x + - 0)
H3:0 < x0 < x
- exp 0 + (1 + x) < exp x0 * x
x:R
H:0 < x
H0:exists c : R,
  exp x - exp 0 =
  derive_pt exp c (derivable_exp c) * (x - 0) /\
  0 < c < x
x0:R
H1:exp x - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x - 0) /\
0 < x0 < x
H2:exp x + - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x + - 0)
H3:0 < x0 < x
exp x0 = derive_pt exp x0 (derivable_exp x0)
  rewrite exp_0; rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
    pattern x at 1; rewrite <- Rmult_1_r; rewrite (Rmult_comm (exp x0));
      apply Rmult_lt_compat_l.x:R
H:0 < x
H0:exists c : R,
  exp x - exp 0 =
  derive_pt exp c (derivable_exp c) * (x - 0) /\
  0 < c < x
x0:R
H1:exp x - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x - 0) /\
0 < x0 < x
H2:exp x + - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x + - 0)
H3:0 < x0 < x
0 < x
x:R
H:0 < x
H0:exists c : R,
  exp x - exp 0 =
  derive_pt exp c (derivable_exp c) * (x - 0) /\
  0 < c < x
x0:R
H1:exp x - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x - 0) /\
0 < x0 < x
H2:exp x + - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x + - 0)
H3:0 < x0 < x
1 < exp x0
x:R
H:0 < x
H0:exists c : R,
  exp x - exp 0 =
  derive_pt exp c (derivable_exp c) * (x - 0) /\
  0 < c < x
x0:R
H1:exp x - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x - 0) /\
0 < x0 < x
H2:exp x + - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x + - 0)
H3:0 < x0 < x
exp x0 = derive_pt exp x0 (derivable_exp x0)
  apply H.x:R
H:0 < x
H0:exists c : R,
  exp x - exp 0 =
  derive_pt exp c (derivable_exp c) * (x - 0) /\
  0 < c < x
x0:R
H1:exp x - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x - 0) /\
0 < x0 < x
H2:exp x + - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x + - 0)
H3:0 < x0 < x
1 < exp x0
x:R
H:0 < x
H0:exists c : R,
  exp x - exp 0 =
  derive_pt exp c (derivable_exp c) * (x - 0) /\
  0 < c < x
x0:R
H1:exp x - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x - 0) /\
0 < x0 < x
H2:exp x + - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x + - 0)
H3:0 < x0 < x
exp x0 = derive_pt exp x0 (derivable_exp x0)
  rewrite <- exp_0; apply exp_increasing; elim H3; intros; assumption.x:R
H:0 < x
H0:exists c : R,
  exp x - exp 0 =
  derive_pt exp c (derivable_exp c) * (x - 0) /\
  0 < c < x
x0:R
H1:exp x - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x - 0) /\
0 < x0 < x
H2:exp x + - exp 0 =
derive_pt exp x0 (derivable_exp x0) * (x + - 0)
H3:0 < x0 < x
exp x0 = derive_pt exp x0 (derivable_exp x0)
  symmetry ; apply derive_pt_eq_0; apply derivable_pt_lim_exp.
Qed.

Lemma ln_exists1 : forall y:R, 1 <= y -> { z:R | y = exp z }.forall y : R, 1 <= y -> {z : R | y = exp z}
Proof.forall y : R, 1 <= y -> {z : R | y = exp z}
  intros; set (f := fun x:R => exp x - y).y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
{z : R | y = exp z}
  assert (H0 : 0 < y) by (apply Rlt_le_trans with 1; auto with real).y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
{z : R | y = exp z}
  cut (f 0 <= 0); [intro H1|].y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
{z : R | y = exp z}
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
f 0 <= 0
  cut (continuity f); [intro H2|].y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
H2:continuity f
{z : R | y = exp z}
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
continuity f
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
f 0 <= 0
  cut (0 <= f y); [intro H3|].y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
H2:continuity f
H3:0 <= f y
{z : R | y = exp z}
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
H2:continuity f
0 <= f y
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
continuity f
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
f 0 <= 0
  cut (f 0 * f y <= 0); [intro H4|].y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
H2:continuity f
H3:0 <= f y
H4:f 0 * f y <= 0
{z : R | y = exp z}
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
H2:continuity f
H3:0 <= f y
f 0 * f y <= 0
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
H2:continuity f
0 <= f y
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
continuity f
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
f 0 <= 0
  pose proof (IVT_cor f 0 y H2 (Rlt_le _ _ H0) H4) as (t,(_,H7));
    exists t; unfold f in H7; apply Rminus_diag_uniq_sym; exact H7.y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
H2:continuity f
H3:0 <= f y
f 0 * f y <= 0
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
H2:continuity f
0 <= f y
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
continuity f
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
f 0 <= 0
  pattern 0 at 2; rewrite <- (Rmult_0_r (f y));
    rewrite (Rmult_comm (f 0)); apply Rmult_le_compat_l;
      assumption.y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
H2:continuity f
0 <= f y
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
continuity f
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
f 0 <= 0
  unfold f; apply Rplus_le_reg_l with y; left;
    apply Rlt_trans with (1 + y).y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
H2:continuity f
y + 0 < 1 + y
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
H2:continuity f
1 + y < y + (exp y - y)
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
continuity f
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
f 0 <= 0
  rewrite <- (Rplus_comm y); apply Rplus_lt_compat_l; apply Rlt_0_1.y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
H2:continuity f
1 + y < y + (exp y - y)
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
continuity f
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
f 0 <= 0
  replace (y + (exp y - y)) with (exp y); [ apply (exp_ineq1 y H0) | ring ].y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
H1:f 0 <= 0
continuity f
y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
f 0 <= 0
  unfold f; change (continuity (exp - fct_cte y));
    apply continuity_minus;
      [ apply derivable_continuous; apply derivable_exp
        | apply derivable_continuous; apply derivable_const ].y:R
H:1 <= y
f:=fun x : R => exp x - y:R -> R
H0:0 < y
f 0 <= 0
  unfold f; rewrite exp_0; apply Rplus_le_reg_l with y;
    rewrite Rplus_0_r; replace (y + (1 - y)) with 1; [ apply H | ring ].
Qed.

(**********)
Lemma ln_exists : forall y:R, 0 < y -> { z:R | y = exp z }.forall y : R, 0 < y -> {z : R | y = exp z}
Proof.forall y : R, 0 < y -> {z : R | y = exp z}
  intros; destruct (Rle_dec 1 y) as [Hle|Hnle].y:R
H:0 < y
Hle:1 <= y
{z : R | y = exp z}
y:R
H:0 < y
Hnle:~ 1 <= y
{z : R | y = exp z}
  apply (ln_exists1 _ Hle).y:R
H:0 < y
Hnle:~ 1 <= y
{z : R | y = exp z}
  assert (H0 : 1 <= / y).y:R
H:0 < y
Hnle:~ 1 <= y
1 <= / y
y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
{z : R | y = exp z}
  apply Rmult_le_reg_l with y.y:R
H:0 < y
Hnle:~ 1 <= y
0 < y
y:R
H:0 < y
Hnle:~ 1 <= y
y * 1 <= y * / y
y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
{z : R | y = exp z}
  apply H.y:R
H:0 < y
Hnle:~ 1 <= y
y * 1 <= y * / y
y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
{z : R | y = exp z}
  rewrite <- Rinv_r_sym.y:R
H:0 < y
Hnle:~ 1 <= y
y * 1 <= 1
y:R
H:0 < y
Hnle:~ 1 <= y
y <> 0
y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
{z : R | y = exp z}
  rewrite Rmult_1_r; left; apply (Rnot_le_lt _ _ Hnle).y:R
H:0 < y
Hnle:~ 1 <= y
y <> 0
y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
{z : R | y = exp z}
  red; intro; rewrite H0 in H; elim (Rlt_irrefl _ H).y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
{z : R | y = exp z}
  destruct (ln_exists1 _ H0) as (x,p); exists (- x);
    apply Rmult_eq_reg_l with (exp x / y).y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
x:R
p:/ y = exp x
exp x / y * y = exp x / y * exp (- x)
y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
x:R
p:/ y = exp x
exp x / y <> 0
  unfold Rdiv; rewrite Rmult_assoc; rewrite <- Rinv_l_sym.y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
x:R
p:/ y = exp x
exp x * 1 = exp x * / y * exp (- x)
y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
x:R
p:/ y = exp x
y <> 0
y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
x:R
p:/ y = exp x
exp x / y <> 0
  rewrite Rmult_1_r; rewrite <- (Rmult_comm (/ y)); rewrite Rmult_assoc;
    rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0;
      rewrite Rmult_1_r; symmetry ; apply p.y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
x:R
p:/ y = exp x
y <> 0
y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
x:R
p:/ y = exp x
exp x / y <> 0
  red; intro H3; rewrite H3 in H; elim (Rlt_irrefl _ H).y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
x:R
p:/ y = exp x
exp x / y <> 0
  unfold Rdiv; apply prod_neq_R0.y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
x:R
p:/ y = exp x
exp x <> 0
y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
x:R
p:/ y = exp x
/ y <> 0
  assert (H3 := exp_pos x); red; intro H4; rewrite H4 in H3;
    elim (Rlt_irrefl _ H3).y:R
H:0 < y
Hnle:~ 1 <= y
H0:1 <= / y
x:R
p:/ y = exp x
/ y <> 0
  apply Rinv_neq_0_compat; red; intro H3; rewrite H3 in H;
    elim (Rlt_irrefl _ H).
Qed.

(* Definition of log R+* -> R *)
Definition Rln (y:posreal) : R :=
  let (a,_) := ln_exists (pos y) (cond_pos y) in a.

(* Extension on R *)
Definition ln (x:R) : R :=
  match Rlt_dec 0 x with
    | left a => Rln (mkposreal x a)
    | right a => 0
  end.

Lemma exp_ln : forall x:R, 0 < x -> exp (ln x) = x.forall x : R, 0 < x -> exp (ln x) = x
Proof.forall x : R, 0 < x -> exp (ln x) = x
  intros; unfold ln; decide (Rlt_dec 0 x) with H.x:R
H:0 < x
exp (Rln {| pos := x; cond_pos := H |}) = x
  unfold Rln;
    case (ln_exists (mkposreal x H) (cond_pos (mkposreal x H))) as (?,Hex).x:R
H:0 < x
x0:R
Hex:{| pos := x; cond_pos := H |} = exp x0
exp x0 = x
  symmetry; apply Hex.
Qed.

Theorem exp_inv : forall x y:R, exp x = exp y -> x = y.forall x y : R, exp x = exp y -> x = y
Proof.forall x y : R, exp x = exp y -> x = y
  intros x y H; case (Rtotal_order x y); [ intros H1 | intros [H1| H1] ]; auto;
    assert (H2 := exp_increasing _ _ H1); rewrite H in H2;
      elim (Rlt_irrefl _ H2).
Qed.

Theorem exp_Ropp : forall x:R, exp (- x) = / exp x.forall x : R, exp (- x) = / exp x
Proof.forall x : R, exp (- x) = / exp x
  intros x; assert (H : exp x <> 0).x:R
exp x <> 0
x:R
H:exp x <> 0
exp (- x) = / exp x
  assert (H := exp_pos x); red; intro; rewrite H0 in H;
    elim (Rlt_irrefl _ H).x:R
H:exp x <> 0
exp (- x) = / exp x
  apply Rmult_eq_reg_l with (r := exp x).x:R
H:exp x <> 0
exp x * exp (- x) = exp x * / exp x
x:R
H:exp x <> 0
exp x <> 0
  rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0.x:R
H:exp x <> 0
1 = exp x * / exp x
x:R
H:exp x <> 0
exp x <> 0
  apply Rinv_r_sym.x:R
H:exp x <> 0
exp x <> 0
x:R
H:exp x <> 0
exp x <> 0
  apply H.x:R
H:exp x <> 0
exp x <> 0
  apply H.
Qed.

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

Properties of Ln

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

Theorem ln_increasing : forall x y:R, 0 < x -> x < y -> ln x < ln y.forall x y : R, 0 < x -> x < y -> ln x < ln y
Proof.forall x y : R, 0 < x -> x < y -> ln x < ln y
  intros x y H H0; apply exp_lt_inv.x, y:R
H:0 < x
H0:x < y
exp (ln x) < exp (ln y)
  repeat rewrite exp_ln.x, y:R
H:0 < x
H0:x < y
x < y
x, y:R
H:0 < x
H0:x < y
0 < y
x, y:R
H:0 < x
H0:x < y
0 < x
  apply H0.x, y:R
H:0 < x
H0:x < y
0 < y
x, y:R
H:0 < x
H0:x < y
0 < x
  apply Rlt_trans with x; assumption.x, y:R
H:0 < x
H0:x < y
0 < x
  apply H.
Qed.

Theorem ln_exp : forall x:R, ln (exp x) = x.forall x : R, ln (exp x) = x
Proof.forall x : R, ln (exp x) = x
  intros x; apply exp_inv.x:R
exp (ln (exp x)) = exp x
  apply exp_ln.x:R
0 < exp x
  apply exp_pos.
Qed.

Theorem ln_1 : ln 1 = 0.ln 1 = 0
Proof.ln 1 = 0
  rewrite <- exp_0; rewrite ln_exp; reflexivity.
Qed.

Theorem ln_lt_inv : forall x y:R, 0 < x -> 0 < y -> ln x < ln y -> x < y.forall x y : R, 0 < x -> 0 < y -> ln x < ln y -> x < y
Proof.forall x y : R, 0 < x -> 0 < y -> ln x < ln y -> x < y
  intros x y H H0 H1; rewrite <- (exp_ln x); try rewrite <- (exp_ln y).x, y:R
H:0 < x
H0:0 < y
H1:ln x < ln y
exp (ln x) < exp (ln y)
x, y:R
H:0 < x
H0:0 < y
H1:ln x < ln y
0 < y
x, y:R
H:0 < x
H0:0 < y
H1:ln x < ln y
0 < x
  apply exp_increasing; apply H1.x, y:R
H:0 < x
H0:0 < y
H1:ln x < ln y
0 < y
x, y:R
H:0 < x
H0:0 < y
H1:ln x < ln y
0 < x
  assumption.x, y:R
H:0 < x
H0:0 < y
H1:ln x < ln y
0 < x
  assumption.
Qed.

Theorem ln_inv : forall x y:R, 0 < x -> 0 < y -> ln x = ln y -> x = y.forall x y : R, 0 < x -> 0 < y -> ln x = ln y -> x = y
Proof.forall x y : R, 0 < x -> 0 < y -> ln x = ln y -> x = y
  intros x y H H0 H'0; case (Rtotal_order x y); [ intros H1 | intros [H1| H1] ];
    auto.x, y:R
H:0 < x
H0:0 < y
H'0:ln x = ln y
H1:x < y
x = y
x, y:R
H:0 < x
H0:0 < y
H'0:ln x = ln y
H1:x > y
x = y
  assert (H2 := ln_increasing _ _ H H1); rewrite H'0 in H2;
    elim (Rlt_irrefl _ H2).x, y:R
H:0 < x
H0:0 < y
H'0:ln x = ln y
H1:x > y
x = y
  assert (H2 := ln_increasing _ _ H0 H1); rewrite H'0 in H2;
    elim (Rlt_irrefl _ H2).
Qed.

Theorem ln_mult : forall x y:R, 0 < x -> 0 < y -> ln (x * y) = ln x + ln y.forall x y : R,
0 < x -> 0 < y -> ln (x * y) = ln x + ln y
Proof.forall x y : R,
0 < x -> 0 < y -> ln (x * y) = ln x + ln y
  intros x y H H0; apply exp_inv.x, y:R
H:0 < x
H0:0 < y
exp (ln (x * y)) = exp (ln x + ln y)
  rewrite exp_plus.x, y:R
H:0 < x
H0:0 < y
exp (ln (x * y)) = exp (ln x) * exp (ln y)
  repeat rewrite exp_ln.x, y:R
H:0 < x
H0:0 < y
x * y = x * y
x, y:R
H:0 < x
H0:0 < y
0 < y
x, y:R
H:0 < x
H0:0 < y
0 < x
x, y:R
H:0 < x
H0:0 < y
0 < x * y
  reflexivity.x, y:R
H:0 < x
H0:0 < y
0 < y
x, y:R
H:0 < x
H0:0 < y
0 < x
x, y:R
H:0 < x
H0:0 < y
0 < x * y
  assumption.x, y:R
H:0 < x
H0:0 < y
0 < x
x, y:R
H:0 < x
H0:0 < y
0 < x * y
  assumption.x, y:R
H:0 < x
H0:0 < y
0 < x * y
  apply Rmult_lt_0_compat; assumption.
Qed.

Theorem ln_Rinv : forall x:R, 0 < x -> ln (/ x) = - ln x.forall x : R, 0 < x -> ln (/ x) = - ln x
Proof.forall x : R, 0 < x -> ln (/ x) = - ln x
  intros x H; apply exp_inv; repeat rewrite exp_ln || rewrite exp_Ropp.x:R
H:0 < x
/ x = / x
x:R
H:0 < x
0 < x
x:R
H:0 < x
0 < / x
  reflexivity.x:R
H:0 < x
0 < x
x:R
H:0 < x
0 < / x
  assumption.x:R
H:0 < x
0 < / x
  apply Rinv_0_lt_compat; assumption.
Qed.

Theorem ln_continue :
  forall y:R, 0 < y -> continue_in ln (fun x:R => 0 < x) y.forall y : R,
0 < y -> continue_in ln (fun x : R => 0 < x) y
Proof.forall y : R,
0 < y -> continue_in ln (fun x : R => 0 < x) y
  intros y H.y:R
H:0 < y
continue_in ln (fun x : R => 0 < x) y
  unfold continue_in, limit1_in, limit_in; intros eps Heps.y:R
H:0 < y
eps:R
Heps:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x (fun x0 : R => 0 < x0) y x /\
   dist R_met x y < alp ->
   dist R_met (ln x) (ln y) < eps)
  cut (1 < exp eps); [ intros H1 | idtac ].y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x (fun x0 : R => 0 < x0) y x /\
   dist R_met x y < alp ->
   dist R_met (ln x) (ln y) < eps)
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  cut (exp (- eps) < 1); [ intros H2 | idtac ].y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x (fun x0 : R => 0 < x0) y x /\
   dist R_met x y < alp ->
   dist R_met (ln x) (ln y) < eps)
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  exists (Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))); split.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps))) > 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
forall x : Base R_met,
D_x (fun x0 : R => 0 < x0) y x /\
dist R_met x y <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps))) ->
dist R_met (ln x) (ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  red; apply P_Rmin.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
0 < y * (exp eps - 1)
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
0 < y * (1 - exp (- eps))
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
forall x : Base R_met,
D_x (fun x0 : R => 0 < x0) y x /\
dist R_met x y <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps))) ->
dist R_met (ln x) (ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rmult_lt_0_compat.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
0 < y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
0 < exp eps - 1
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
0 < y * (1 - exp (- eps))
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
forall x : Base R_met,
D_x (fun x0 : R => 0 < x0) y x /\
dist R_met x y <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps))) ->
dist R_met (ln x) (ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  assumption.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
0 < exp eps - 1
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
0 < y * (1 - exp (- eps))
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
forall x : Base R_met,
D_x (fun x0 : R => 0 < x0) y x /\
dist R_met x y <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps))) ->
dist R_met (ln x) (ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rplus_lt_reg_l with 1.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
1 + 0 < 1 + (exp eps - 1)
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
0 < y * (1 - exp (- eps))
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
forall x : Base R_met,
D_x (fun x0 : R => 0 < x0) y x /\
dist R_met x y <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps))) ->
dist R_met (ln x) (ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite Rplus_0_r; replace (1 + (exp eps - 1)) with (exp eps);
    [ apply H1 | ring ].y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
0 < y * (1 - exp (- eps))
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
forall x : Base R_met,
D_x (fun x0 : R => 0 < x0) y x /\
dist R_met x y <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps))) ->
dist R_met (ln x) (ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rmult_lt_0_compat.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
0 < y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
0 < 1 - exp (- eps)
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
forall x : Base R_met,
D_x (fun x0 : R => 0 < x0) y x /\
dist R_met x y <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps))) ->
dist R_met (ln x) (ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  assumption.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
0 < 1 - exp (- eps)
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
forall x : Base R_met,
D_x (fun x0 : R => 0 < x0) y x /\
dist R_met x y <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps))) ->
dist R_met (ln x) (ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rplus_lt_reg_l with (exp (- eps)).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
exp (- eps) + 0 < exp (- eps) + (1 - exp (- eps))
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
forall x : Base R_met,
D_x (fun x0 : R => 0 < x0) y x /\
dist R_met x y <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps))) ->
dist R_met (ln x) (ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite Rplus_0_r; replace (exp (- eps) + (1 - exp (- eps))) with 1;
    [ apply H2 | ring ].y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
forall x : Base R_met,
D_x (fun x0 : R => 0 < x0) y x /\
dist R_met x y <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps))) ->
dist R_met (ln x) (ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  unfold dist, R_met, R_dist; simpl.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
forall x : R,
D_x (fun x0 : R => 0 < x0) y x /\
Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps))) ->
Rabs (ln x - ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  intros x [[H3 H4] H5].y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Rabs (ln x - ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  cut (y * (x * / y) = x).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x -> Rabs (ln x - ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  intro Hxyy.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Rabs (ln x - ln y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  replace (ln x - ln y) with (ln (x * / y)).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  case (Rtotal_order x y); [ intros Hxy | intros [Hxy| Hxy] ].y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite Rabs_left.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
- ln (x * / y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Ropp_lt_cancel; rewrite Ropp_involutive.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
- eps < ln (x * / y)
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply exp_lt_inv.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
exp (- eps) < exp (ln (x * / y))
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite exp_ln.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
exp (- eps) < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rmult_lt_reg_l with (r := y).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
y * exp (- eps) < y * (x * / y)
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply H.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
y * exp (- eps) < y * (x * / y)
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite Hxyy.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
y * exp (- eps) < x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Ropp_lt_cancel.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
- x < - (y * exp (- eps))
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rplus_lt_reg_l with (r := y).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
y + - x < y + - (y * exp (- eps))
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  replace (y + - (y * exp (- eps))) with (y * (1 - exp (- eps)));
  [ idtac | ring ].y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
y + - x < y * (1 - exp (- eps))
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  replace (y + - x) with (Rabs (x - y)).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
Rabs (x - y) < y * (1 - exp (- eps))
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
Rabs (x - y) = y + - x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rlt_le_trans with (1 := H5); apply Rmin_r.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
Rabs (x - y) = y + - x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite Rabs_left; [ ring | idtac ].y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
x - y < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply (Rlt_minus _ _ Hxy).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rmult_lt_0_compat; [ apply H3 | apply (Rinv_0_lt_compat _ H) ].y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite <- ln_1.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
ln (x * / y) < ln 1
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply ln_increasing.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
x * / y < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rmult_lt_0_compat; [ apply H3 | apply (Rinv_0_lt_compat _ H) ].y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
x * / y < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rmult_lt_reg_l with (r := y).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
0 < y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
y * (x * / y) < y * 1
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply H.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x < y
y * (x * / y) < y * 1
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite Hxyy; rewrite Rmult_1_r; apply Hxy.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite Hxy; rewrite Rinv_r.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
Rabs (ln 1) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
y <> 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite ln_1; rewrite Rabs_R0; apply Heps.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x = y
y <> 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  red; intro; rewrite H0 in H; elim (Rlt_irrefl _ H).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (ln (x * / y)) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite Rabs_right.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) < eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply exp_lt_inv.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
exp (ln (x * / y)) < exp eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite exp_ln.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
x * / y < exp eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rmult_lt_reg_l with (r := y).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
0 < y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
y * (x * / y) < y * exp eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply H.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
y * (x * / y) < y * exp eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite Hxyy.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
x < y * exp eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rplus_lt_reg_l with (r := - y).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
- y + x < - y + y * exp eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  replace (- y + y * exp eps) with (y * (exp eps - 1)); [ idtac | ring ].y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
- y + x < y * (exp eps - 1)
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  replace (- y + x) with (Rabs (x - y)).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (x - y) < y * (exp eps - 1)
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (x - y) = - y + x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rlt_le_trans with (1 := H5); apply Rmin_l.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
Rabs (x - y) = - y + x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite Rabs_right; [ ring | idtac ].y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
x - y >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  left; apply (Rgt_minus _ _ Hxy).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
0 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rmult_lt_0_compat; [ apply H3 | apply (Rinv_0_lt_compat _ H) ].y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite <- ln_1.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
ln (x * / y) >= ln 1
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rgt_ge; red; apply ln_increasing.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
0 < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
1 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rlt_0_1.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
1 < x * / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rmult_lt_reg_l with (r := y).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
0 < y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
y * 1 < y * (x * / y)
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply H.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
Hxy:x > y
y * 1 < y * (x * / y)
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite Hxyy; rewrite Rmult_1_r; apply Hxy.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln (x * / y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite ln_mult.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln x + ln (/ y) = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
0 < x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
0 < / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite ln_Rinv.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
ln x + - ln y = ln x - ln y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
0 < y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
0 < x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
0 < / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  ring.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
0 < y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
0 < x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
0 < / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  assumption.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
0 < x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
0 < / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  assumption.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
Hxyy:y * (x * / y) = x
0 < / y
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rinv_0_lt_compat; assumption.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y * (x * / y) = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite (Rmult_comm x); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
1 * x = x
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y <> 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  ring.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
H2:exp (- eps) < 1
x:R
H3:0 < x
H4:y <> x
H5:Rabs (x - y) <
Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))
y <> 0
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  red; intro; rewrite H0 in H; elim (Rlt_irrefl _ H).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp (- eps) < 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply Rmult_lt_reg_l with (exp eps).y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
0 < exp eps
y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp eps * exp (- eps) < exp eps * 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  apply exp_pos.y:R
H:0 < y
eps:R
Heps:eps > 0
H1:1 < exp eps
exp eps * exp (- eps) < exp eps * 1
y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite <- exp_plus; rewrite Rmult_1_r; rewrite Rplus_opp_r; rewrite exp_0;
    apply H1.y:R
H:0 < y
eps:R
Heps:eps > 0
1 < exp eps
  rewrite <- exp_0.y:R
H:0 < y
eps:R
Heps:eps > 0
exp 0 < exp eps
  apply exp_increasing; apply Heps.
Qed.

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

Definition of Rpower

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

Definition Rpower (x y:R) := exp (y * ln x).

Local Infix "^R" := Rpower (at level 30, right associativity) : R_scope.

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

Properties of Rpower

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

Note: Rpower is prolongated to 1 on negative real numbers and it thus does not extend integer power. The next two lemmas, which hold for integer power, accidentally hold on negative real numbers as a side effect of the default value taken on negative real numbers. Contrastingly, the lemmas that do not hold for the integer power of a negative number are stated for Rpower on the positive numbers only (even if they accidentally hold due to the default value of Rpower on the negative side, as it is the case for Rpower_O).

Theorem Rpower_plus : forall x y z:R, z ^R (x + y) = z ^R x * z ^R y.forall x y z : R, z ^R (x + y) = z ^R x * z ^R y
Proof.forall x y z : R, z ^R (x + y) = z ^R x * z ^R y
  intros x y z; unfold Rpower.x, y, z:R
exp ((x + y) * ln z) = exp (x * ln z) * exp (y * ln z)
  rewrite Rmult_plus_distr_r; rewrite exp_plus; auto.
Qed.

Theorem Rpower_mult : forall x y z:R, (x ^R y) ^R z = x ^R (y * z).forall x y z : R, (x ^R y) ^R z = x ^R (y * z)
Proof.forall x y z : R, (x ^R y) ^R z = x ^R (y * z)
  intros x y z; unfold Rpower.x, y, z:R
exp (z * ln (exp (y * ln x))) = exp (y * z * ln x)
  rewrite ln_exp.x, y, z:R
exp (z * (y * ln x)) = exp (y * z * ln x)
  replace (z * (y * ln x)) with (y * z * ln x).x, y, z:R
exp (y * z * ln x) = exp (y * z * ln x)
x, y, z:R
y * z * ln x = z * (y * ln x)
  reflexivity.x, y, z:R
y * z * ln x = z * (y * ln x)
  ring.
Qed.

Theorem Rpower_O : forall x:R, 0 < x -> x ^R 0 = 1.forall x : R, 0 < x -> x ^R 0 = 1
Proof.forall x : R, 0 < x -> x ^R 0 = 1
  intros x _; unfold Rpower.x:R
exp (0 * ln x) = 1
  rewrite Rmult_0_l; apply exp_0.
Qed.

Theorem Rpower_1 : forall x:R, 0 < x -> x ^R 1 = x.forall x : R, 0 < x -> x ^R 1 = x
Proof.forall x : R, 0 < x -> x ^R 1 = x
  intros x H; unfold Rpower.x:R
H:0 < x
exp (1 * ln x) = x
  rewrite Rmult_1_l; apply exp_ln; apply H.
Qed.

Theorem Rpower_pow : forall (n:nat) (x:R), 0 < x -> x ^R INR n = x ^ n.forall (n : nat) (x : R), 0 < x -> x ^R INR n = x ^ n
Proof.forall (n : nat) (x : R), 0 < x -> x ^R INR n = x ^ n
  intros n; elim n; simpl; auto; fold INR.n:nat
forall x : R, 0 < x -> x ^R 0 = 1
n:nat
forall n0 : nat,
(forall x : R, 0 < x -> x ^R INR n0 = x ^ n0) ->
forall x : R,
0 < x ->
x
^R match n0 with
   | 0%nat => 1
   | S _ => INR n0 + 1
   end = x * x ^ n0
  intros x H; apply Rpower_O; auto.n:nat
forall n0 : nat,
(forall x : R, 0 < x -> x ^R INR n0 = x ^ n0) ->
forall x : R,
0 < x ->
x
^R match n0 with
   | 0%nat => 1
   | S _ => INR n0 + 1
   end = x * x ^ n0
  intros n1; case n1.n, n1:nat
(forall x : R, 0 < x -> x ^R INR 0 = x ^ 0) ->
forall x : R, 0 < x -> x ^R 1 = x * x ^ 0
n, n1:nat
forall n0 : nat,
(forall x : R, 0 < x -> x ^R INR (S n0) = x ^ S n0) ->
forall x : R,
0 < x -> x ^R (INR (S n0) + 1) = x * x ^ S n0
  intros H x H0; simpl; rewrite Rmult_1_r; apply Rpower_1; auto.n, n1:nat
forall n0 : nat,
(forall x : R, 0 < x -> x ^R INR (S n0) = x ^ S n0) ->
forall x : R,
0 < x -> x ^R (INR (S n0) + 1) = x * x ^ S n0
  intros n0 H x H0; rewrite Rpower_plus; rewrite H; try rewrite Rpower_1;
    try apply Rmult_comm || assumption.
Qed.

Theorem Rpower_lt :
  forall x y z:R, 1 < x -> y < z -> x ^R y < x ^R z.forall x y z : R, 1 < x -> y < z -> x ^R y < x ^R z
Proof.forall x y z : R, 1 < x -> y < z -> x ^R y < x ^R z
  intros x y z H H1.x, y, z:R
H:1 < x
H1:y < z
x ^R y < x ^R z
  unfold Rpower.x, y, z:R
H:1 < x
H1:y < z
exp (y * ln x) < exp (z * ln x)
  apply exp_increasing.x, y, z:R
H:1 < x
H1:y < z
y * ln x < z * ln x
  apply Rmult_lt_compat_r.x, y, z:R
H:1 < x
H1:y < z
0 < ln x
x, y, z:R
H:1 < x
H1:y < z
y < z
  rewrite <- ln_1; apply ln_increasing.x, y, z:R
H:1 < x
H1:y < z
0 < 1
x, y, z:R
H:1 < x
H1:y < z
1 < x
x, y, z:R
H:1 < x
H1:y < z
y < z
  apply Rlt_0_1.x, y, z:R
H:1 < x
H1:y < z
1 < x
x, y, z:R
H:1 < x
H1:y < z
y < z
  apply H.x, y, z:R
H:1 < x
H1:y < z
y < z
  apply H1.
Qed.

Theorem Rpower_sqrt : forall x:R, 0 < x -> x ^R (/ 2) = sqrt x.forall x : R, 0 < x -> x ^R (/ 2) = sqrt x
Proof.forall x : R, 0 < x -> x ^R (/ 2) = sqrt x
  intros x H.x:R
H:0 < x
x ^R (/ 2) = sqrt x
  apply ln_inv.x:R
H:0 < x
0 < x ^R (/ 2)
x:R
H:0 < x
0 < sqrt x
x:R
H:0 < x
ln (x ^R (/ 2)) = ln (sqrt x)
  unfold Rpower; apply exp_pos.x:R
H:0 < x
0 < sqrt x
x:R
H:0 < x
ln (x ^R (/ 2)) = ln (sqrt x)
  apply sqrt_lt_R0; apply H.x:R
H:0 < x
ln (x ^R (/ 2)) = ln (sqrt x)
  apply Rmult_eq_reg_l with (INR 2).x:R
H:0 < x
INR 2 * ln (x ^R (/ 2)) = INR 2 * ln (sqrt x)
x:R
H:0 < x
INR 2 <> 0
  apply exp_inv.x:R
H:0 < x
exp (INR 2 * ln (x ^R (/ 2))) =
exp (INR 2 * ln (sqrt x))
x:R
H:0 < x
INR 2 <> 0
  fold Rpower.x:R
H:0 < x
exp (INR 2 * ln (x ^R (/ 2))) =
exp (INR 2 * ln (sqrt x))
x:R
H:0 < x
INR 2 <> 0
  cut ((x ^R (/ INR 2)) ^R INR 2 = sqrt x ^R INR 2).x:R
H:0 < x
(x ^R (/ INR 2)) ^R INR 2 = sqrt x ^R INR 2 ->
exp (INR 2 * ln (x ^R (/ 2))) =
exp (INR 2 * ln (sqrt x))
x:R
H:0 < x
(x ^R (/ INR 2)) ^R INR 2 = sqrt x ^R INR 2
x:R
H:0 < x
INR 2 <> 0
  unfold Rpower; auto.x:R
H:0 < x
(x ^R (/ INR 2)) ^R INR 2 = sqrt x ^R INR 2
x:R
H:0 < x
INR 2 <> 0
  rewrite Rpower_mult.x:R
H:0 < x
x ^R (/ INR 2 * INR 2) = sqrt x ^R INR 2
x:R
H:0 < x
INR 2 <> 0
  rewrite Rinv_l.x:R
H:0 < x
x ^R 1 = sqrt x ^R INR 2
x:R
H:0 < x
INR 2 <> 0
x:R
H:0 < x
INR 2 <> 0
  change 1 with (INR 1).x:R
H:0 < x
x ^R INR 1 = sqrt x ^R INR 2
x:R
H:0 < x
INR 2 <> 0
x:R
H:0 < x
INR 2 <> 0
  repeat rewrite Rpower_pow; simpl.x:R
H:0 < x
x * 1 = sqrt x * (sqrt x * 1)
x:R
H:0 < x
0 < sqrt x
x:R
H:0 < x
0 < x
x:R
H:0 < x
INR 2 <> 0
x:R
H:0 < x
INR 2 <> 0
  pattern x at 1; rewrite <- (sqrt_sqrt x (Rlt_le _ _ H)).x:R
H:0 < x
sqrt x * sqrt x * 1 = sqrt x * (sqrt x * 1)
x:R
H:0 < x
0 < sqrt x
x:R
H:0 < x
0 < x
x:R
H:0 < x
INR 2 <> 0
x:R
H:0 < x
INR 2 <> 0
  ring.x:R
H:0 < x
0 < sqrt x
x:R
H:0 < x
0 < x
x:R
H:0 < x
INR 2 <> 0
x:R
H:0 < x
INR 2 <> 0
  apply sqrt_lt_R0; apply H.x:R
H:0 < x
0 < x
x:R
H:0 < x
INR 2 <> 0
x:R
H:0 < x
INR 2 <> 0
  apply H.x:R
H:0 < x
INR 2 <> 0
x:R
H:0 < x
INR 2 <> 0
  apply not_O_INR; discriminate.x:R
H:0 < x
INR 2 <> 0
  apply not_O_INR; discriminate.
Qed.

Theorem Rpower_Ropp : forall x y:R, x ^R (- y) = / x ^R y.forall x y : R, x ^R (- y) = / x ^R y
Proof.forall x y : R, x ^R (- y) = / x ^R y
  unfold Rpower.forall x y : R, exp (- y * ln x) = / exp (y * ln x)
  intros x y; rewrite Ropp_mult_distr_l_reverse.x, y:R
exp (- (y * ln x)) = / exp (y * ln x)
  apply exp_Ropp.
Qed.

Lemma powerRZ_Rpower x z : (0 < x)%R -> powerRZ x z = Rpower x (IZR z).x:R
z:Z
0 < x -> powerRZ x z = x ^R IZR z
Proof.x:R
z:Z
0 < x -> powerRZ x z = x ^R IZR z
  intros Hx.x:R
z:Z
Hx:0 < x
powerRZ x z = x ^R IZR z
  assert (x <> 0)%R
    by now intros Habs; rewrite Habs in Hx; apply (Rlt_irrefl 0).x:R
z:Z
Hx:0 < x
H:x <> 0
powerRZ x z = x ^R IZR z
  destruct (intP z).x:R
z:Z
Hx:0 < x
H:x <> 0
e:z = 0%Z
powerRZ x 0 = x ^R 0
x:R
z:Z
Hx:0 < x
H:x <> 0
n:nat
e:z = Z.of_nat n
powerRZ x (Z.of_nat n) = x ^R IZR (Z.of_nat n)
x:R
z:Z
Hx:0 < x
H:x <> 0
n:nat
e:z = (- Z.of_nat n)%Z
powerRZ x (- Z.of_nat n) = x ^R IZR (- Z.of_nat n)
  -x:R
z:Z
Hx:0 < x
H:x <> 0
e:z = 0%Z
powerRZ x 0 = x ^R 0 now rewrite Rpower_O.
  -x:R
z:Z
Hx:0 < x
H:x <> 0
n:nat
e:z = Z.of_nat n
powerRZ x (Z.of_nat n) = x ^R IZR (Z.of_nat n) rewrite <- pow_powerRZ, <- Rpower_pow by assumption.x:R
z:Z
Hx:0 < x
H:x <> 0
n:nat
e:z = Z.of_nat n
x ^R INR n = x ^R IZR (Z.of_nat n)
    now rewrite INR_IZR_INZ.
  -x:R
z:Z
Hx:0 < x
H:x <> 0
n:nat
e:z = (- Z.of_nat n)%Z
powerRZ x (- Z.of_nat n) = x ^R IZR (- Z.of_nat n) rewrite opp_IZR, Rpower_Ropp.x:R
z:Z
Hx:0 < x
H:x <> 0
n:nat
e:z = (- Z.of_nat n)%Z
powerRZ x (- Z.of_nat n) = / x ^R IZR (Z.of_nat n)
    rewrite powerRZ_neg, powerRZ_inv by assumption.x:R
z:Z
Hx:0 < x
H:x <> 0
n:nat
e:z = (- Z.of_nat n)%Z
/ powerRZ x (Z.of_nat n) = / x ^R IZR (Z.of_nat n)
    now rewrite <- pow_powerRZ, <- INR_IZR_INZ, Rpower_pow.
Qed.

Theorem Rle_Rpower :
  forall e n m:R, 1 <= e -> n <= m -> e ^R n <= e ^R m.forall e n m : R, 1 <= e -> n <= m -> e ^R n <= e ^R m
Proof.forall e n m : R, 1 <= e -> n <= m -> e ^R n <= e ^R m
  intros e n m [H | H]; intros H1.e, n, m:R
H:1 < e
H1:n <= m
e ^R n <= e ^R m
e, n, m:R
H:1 = e
H1:n <= m
e ^R n <= e ^R m
  case H1.e, n, m:R
H:1 < e
H1:n <= m
n < m -> e ^R n <= e ^R m
e, n, m:R
H:1 < e
H1:n <= m
n = m -> e ^R n <= e ^R m
e, n, m:R
H:1 = e
H1:n <= m
e ^R n <= e ^R m
    intros H2; left; apply Rpower_lt; assumption.e, n, m:R
H:1 < e
H1:n <= m
n = m -> e ^R n <= e ^R m
e, n, m:R
H:1 = e
H1:n <= m
e ^R n <= e ^R m
    intros H2; rewrite H2; right; reflexivity.e, n, m:R
H:1 = e
H1:n <= m
e ^R n <= e ^R m
  now rewrite <- H; unfold Rpower; rewrite ln_1, !Rmult_0_r; apply Rle_refl.
Qed.

Theorem ln_lt_2 : / 2 < ln 2./ 2 < ln 2
Proof./ 2 < ln 2
  apply Rmult_lt_reg_l with (r := 2).0 < 2
2 * / 2 < 2 * ln 2
  prove_sup0.2 * / 2 < 2 * ln 2
  rewrite Rinv_r.1 < 2 * ln 2
2 <> 0
  apply exp_lt_inv.exp 1 < exp (2 * ln 2)
2 <> 0
  apply Rle_lt_trans with (1 := exp_le_3).3 < exp (2 * ln 2)
2 <> 0
  change (3 < 2 ^R (1 + 1)).3 < 2 ^R (1 + 1)
2 <> 0
  repeat rewrite Rpower_plus; repeat rewrite Rpower_1.3 < 2 * 2
0 < 2
2 <> 0
  now apply (IZR_lt 3 4).0 < 2
2 <> 0
  prove_sup0.2 <> 0
  discrR.
Qed.

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

Differentiability of Ln and Rpower

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

Theorem limit1_ext :
  forall (f g:R -> R) (D:R -> Prop) (l x:R),
    (forall x:R, D x -> f x = g x) -> limit1_in f D l x -> limit1_in g D l x.forall (f g : R -> R) (D : R -> Prop) (l x : R),
(forall x0 : R, D x0 -> f x0 = g x0) ->
limit1_in f D l x -> limit1_in g D l x
Proof.forall (f g : R -> R) (D : R -> Prop) (l x : R),
(forall x0 : R, D x0 -> f x0 = g x0) ->
limit1_in f D l x -> limit1_in g D l x
  intros f g D l x H; unfold limit1_in, limit_in.f, g:R -> R
D:R -> Prop
l, x:R
H:forall x0 : R, D x0 -> f x0 = g x0
(forall eps : R,
 eps > 0 ->
 exists alp : R,
   alp > 0 /\
   (forall x0 : Base R_met,
    D x0 /\ dist R_met x0 x < alp ->
    dist R_met (f x0) l < eps)) ->
forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D x0 /\ dist R_met x0 x < alp ->
   dist R_met (g x0) l < eps)
  intros H0 eps H1; case (H0 eps); auto.f, g:R -> R
D:R -> Prop
l, x:R
H:forall x0 : R, D x0 -> f x0 = g x0
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D x0 /\ dist R_met x0 x < alp ->
   dist R_met (f x0) l < eps0)
eps:R
H1:eps > 0
forall x0 : R,
x0 > 0 /\
(forall x1 : Base R_met,
 D x1 /\ dist R_met x1 x < x0 ->
 dist R_met (f x1) l < eps) ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D x1 /\ dist R_met x1 x < alp ->
   dist R_met (g x1) l < eps)
  intros x0 [H2 H3]; exists x0; split; auto.f, g:R -> R
D:R -> Prop
l, x:R
H:forall x1 : R, D x1 -> f x1 = g x1
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D x1 /\ dist R_met x1 x < alp ->
   dist R_met (f x1) l < eps0)
eps:R
H1:eps > 0
x0:R
H2:x0 > 0
H3:forall x1 : Base R_met,
D x1 /\ dist R_met x1 x < x0 ->
dist R_met (f x1) l < eps
forall x1 : Base R_met,
D x1 /\ dist R_met x1 x < x0 ->
dist R_met (g x1) l < eps
  intros x1 [H4 H5]; rewrite <- H; auto.
Qed.

Theorem limit1_imp :
  forall (f:R -> R) (D D1:R -> Prop) (l x:R),
    (forall x:R, D1 x -> D x) -> limit1_in f D l x -> limit1_in f D1 l x.forall (f : R -> R) (D D1 : R -> Prop) (l x : R),
(forall x0 : R, D1 x0 -> D x0) ->
limit1_in f D l x -> limit1_in f D1 l x
Proof.forall (f : R -> R) (D D1 : R -> Prop) (l x : R),
(forall x0 : R, D1 x0 -> D x0) ->
limit1_in f D l x -> limit1_in f D1 l x
  intros f D D1 l x H; unfold limit1_in, limit_in.f:R -> R
D, D1:R -> Prop
l, x:R
H:forall x0 : R, D1 x0 -> D x0
(forall eps : R,
 eps > 0 ->
 exists alp : R,
   alp > 0 /\
   (forall x0 : Base R_met,
    D x0 /\ dist R_met x0 x < alp ->
    dist R_met (f x0) l < eps)) ->
forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D1 x0 /\ dist R_met x0 x < alp ->
   dist R_met (f x0) l < eps)
  intros H0 eps H1; case (H0 eps H1); auto.f:R -> R
D, D1:R -> Prop
l, x:R
H:forall x0 : R, D1 x0 -> D x0
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D x0 /\ dist R_met x0 x < alp ->
   dist R_met (f x0) l < eps0)
eps:R
H1:eps > 0
forall x0 : R,
x0 > 0 /\
(forall x1 : Base R_met,
 D x1 /\ dist R_met x1 x < x0 ->
 dist R_met (f x1) l < eps) ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D1 x1 /\ dist R_met x1 x < alp ->
   dist R_met (f x1) l < eps)
  intros alpha [H2 H3]; exists alpha; split; auto.f:R -> R
D, D1:R -> Prop
l, x:R
H:forall x0 : R, D1 x0 -> D x0
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D x0 /\ dist R_met x0 x < alp ->
   dist R_met (f x0) l < eps0)
eps:R
H1:eps > 0
alpha:R
H2:alpha > 0
H3:forall x0 : Base R_met,
D x0 /\ dist R_met x0 x < alpha ->
dist R_met (f x0) l < eps
forall x0 : Base R_met,
D1 x0 /\ dist R_met x0 x < alpha ->
dist R_met (f x0) l < eps
  intros d [H4 H5]; apply H3; split; auto.
Qed.

Theorem Rinv_Rdiv : forall x y:R, x <> 0 -> y <> 0 -> / (x / y) = y / x.forall x y : R, x <> 0 -> y <> 0 -> / (x / y) = y / x
Proof.forall x y : R, x <> 0 -> y <> 0 -> / (x / y) = y / x
  intros x y H1 H2; unfold Rdiv; rewrite Rinv_mult_distr.x, y:R
H1:x <> 0
H2:y <> 0
/ x * / / y = y * / x
x, y:R
H1:x <> 0
H2:y <> 0
x <> 0
x, y:R
H1:x <> 0
H2:y <> 0
/ y <> 0
  rewrite Rinv_involutive.x, y:R
H1:x <> 0
H2:y <> 0
/ x * y = y * / x
x, y:R
H1:x <> 0
H2:y <> 0
y <> 0
x, y:R
H1:x <> 0
H2:y <> 0
x <> 0
x, y:R
H1:x <> 0
H2:y <> 0
/ y <> 0
  apply Rmult_comm.x, y:R
H1:x <> 0
H2:y <> 0
y <> 0
x, y:R
H1:x <> 0
H2:y <> 0
x <> 0
x, y:R
H1:x <> 0
H2:y <> 0
/ y <> 0
  assumption.x, y:R
H1:x <> 0
H2:y <> 0
x <> 0
x, y:R
H1:x <> 0
H2:y <> 0
/ y <> 0
  assumption.x, y:R
H1:x <> 0
H2:y <> 0
/ y <> 0
  apply Rinv_neq_0_compat; assumption.
Qed.

Theorem Dln : forall y:R, 0 < y -> D_in ln Rinv (fun x:R => 0 < x) y.forall y : R,
0 < y -> D_in ln Rinv (fun x : R => 0 < x) y
Proof.forall y : R,
0 < y -> D_in ln Rinv (fun x : R => 0 < x) y
  intros y Hy; unfold D_in.y:R
Hy:0 < y
limit1_in (fun x : R => (ln x - ln y) / (x - y))
  (D_x (fun x : R => 0 < x) y) (/ y) y
  apply limit1_ext with
    (f := fun x:R => / ((exp (ln x) - exp (ln y)) / (ln x - ln y))).y:R
Hy:0 < y
forall x : R,
D_x (fun x0 : R => 0 < x0) y x ->
/ ((exp (ln x) - exp (ln y)) / (ln x - ln y)) =
(ln x - ln y) / (x - y)
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   / ((exp (ln x) - exp (ln y)) / (ln x - ln y)))
  (D_x (fun x : R => 0 < x) y) (/ y) y
  intros x [HD1 HD2]; repeat rewrite exp_ln.y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
/ ((x - y) / (ln x - ln y)) = (ln x - ln y) / (x - y)
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < y
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < x
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   / ((exp (ln x) - exp (ln y)) / (ln x - ln y)))
  (D_x (fun x : R => 0 < x) y) (/ y) y
  unfold Rdiv; rewrite Rinv_mult_distr.y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
/ (x - y) * / / (ln x - ln y) =
(ln x - ln y) * / (x - y)
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
x - y <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
/ (ln x - ln y) <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < y
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < x
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   / ((exp (ln x) - exp (ln y)) / (ln x - ln y)))
  (D_x (fun x : R => 0 < x) y) (/ y) y
  rewrite Rinv_involutive.y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
/ (x - y) * (ln x - ln y) = (ln x - ln y) * / (x - y)
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
ln x - ln y <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
x - y <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
/ (ln x - ln y) <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < y
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < x
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   / ((exp (ln x) - exp (ln y)) / (ln x - ln y)))
  (D_x (fun x : R => 0 < x) y) (/ y) y
  apply Rmult_comm.y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
ln x - ln y <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
x - y <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
/ (ln x - ln y) <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < y
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < x
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   / ((exp (ln x) - exp (ln y)) / (ln x - ln y)))
  (D_x (fun x : R => 0 < x) y) (/ y) y
  apply Rminus_eq_contra.y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
ln x <> ln y
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
x - y <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
/ (ln x - ln y) <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < y
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < x
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   / ((exp (ln x) - exp (ln y)) / (ln x - ln y)))
  (D_x (fun x : R => 0 < x) y) (/ y) y
  red; intros H2; case HD2.y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
H2:ln x = ln y
y = x
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
x - y <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
/ (ln x - ln y) <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < y
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < x
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   / ((exp (ln x) - exp (ln y)) / (ln x - ln y)))
  (D_x (fun x : R => 0 < x) y) (/ y) y
  symmetry ; apply (ln_inv _ _ HD1 Hy H2).y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
x - y <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
/ (ln x - ln y) <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < y
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < x
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   / ((exp (ln x) - exp (ln y)) / (ln x - ln y)))
  (D_x (fun x : R => 0 < x) y) (/ y) y
  apply Rminus_eq_contra; apply (not_eq_sym HD2).y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
/ (ln x - ln y) <> 0
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < y
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < x
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   / ((exp (ln x) - exp (ln y)) / (ln x - ln y)))
  (D_x (fun x : R => 0 < x) y) (/ y) y
  apply Rinv_neq_0_compat; apply Rminus_eq_contra; red; intros H2;
    case HD2; apply ln_inv; auto.y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < y
y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < x
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   / ((exp (ln x) - exp (ln y)) / (ln x - ln y)))
  (D_x (fun x : R => 0 < x) y) (/ y) y
  assumption.y:R
Hy:0 < y
x:R
HD1:0 < x
HD2:y <> x
0 < x
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   / ((exp (ln x) - exp (ln y)) / (ln x - ln y)))
  (D_x (fun x : R => 0 < x) y) (/ y) y
  assumption.y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   / ((exp (ln x) - exp (ln y)) / (ln x - ln y)))
  (D_x (fun x : R => 0 < x) y) (/ y) y
  apply limit_inv with
    (f := fun x:R => (exp (ln x) - exp (ln y)) / (ln x - ln y)).y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   (exp (ln x) - exp (ln y)) / (ln x - ln y))
  (D_x (fun x : R => 0 < x) y) y y
y:R
Hy:0 < y
y <> 0
  apply limit1_imp with
    (f := fun x:R => (fun x:R => (exp x - exp (ln y)) / (x - ln y)) (ln x))
    (D := Dgf (D_x (fun x:R => 0 < x) y) (D_x (fun x:R => True) (ln y)) ln).y:R
Hy:0 < y
forall x : R,
D_x (fun x0 : R => 0 < x0) y x ->
Dgf (D_x (fun x0 : R => 0 < x0) y)
  (D_x (fun _ : R => True) (ln y)) ln x
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   (exp (ln x) - exp (ln y)) / (ln x - ln y))
  (Dgf (D_x (fun x : R => 0 < x) y)
     (D_x (fun _ : R => True) (ln y)) ln) y y
y:R
Hy:0 < y
y <> 0
  intros x [H1 H2]; split.y:R
Hy:0 < y
x:R
H1:0 < x
H2:y <> x
D_x (fun x0 : R => 0 < x0) y x
y:R
Hy:0 < y
x:R
H1:0 < x
H2:y <> x
D_x (fun _ : R => True) (ln y) (ln x)
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   (exp (ln x) - exp (ln y)) / (ln x - ln y))
  (Dgf (D_x (fun x : R => 0 < x) y)
     (D_x (fun _ : R => True) (ln y)) ln) y y
y:R
Hy:0 < y
y <> 0
  split; auto.y:R
Hy:0 < y
x:R
H1:0 < x
H2:y <> x
D_x (fun _ : R => True) (ln y) (ln x)
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   (exp (ln x) - exp (ln y)) / (ln x - ln y))
  (Dgf (D_x (fun x : R => 0 < x) y)
     (D_x (fun _ : R => True) (ln y)) ln) y y
y:R
Hy:0 < y
y <> 0
  split; auto.y:R
Hy:0 < y
x:R
H1:0 < x
H2:y <> x
ln y <> ln x
y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   (exp (ln x) - exp (ln y)) / (ln x - ln y))
  (Dgf (D_x (fun x : R => 0 < x) y)
     (D_x (fun _ : R => True) (ln y)) ln) y y
y:R
Hy:0 < y
y <> 0
  red; intros H3; case H2; apply ln_inv; auto.y:R
Hy:0 < y
limit1_in
  (fun x : R =>
   (exp (ln x) - exp (ln y)) / (ln x - ln y))
  (Dgf (D_x (fun x : R => 0 < x) y)
     (D_x (fun _ : R => True) (ln y)) ln) y y
y:R
Hy:0 < y
y <> 0
  apply limit_comp with
    (l := ln y) (g := fun x:R => (exp x - exp (ln y)) / (x - ln y)) (f := ln).y:R
Hy:0 < y
limit1_in ln (D_x (fun x : R => 0 < x) y) (ln y) y
y:R
Hy:0 < y
limit1_in
  (fun x : R => (exp x - exp (ln y)) / (x - ln y))
  (D_x (fun _ : R => True) (ln y)) y (ln y)
y:R
Hy:0 < y
y <> 0
  apply ln_continue; auto.y:R
Hy:0 < y
limit1_in
  (fun x : R => (exp x - exp (ln y)) / (x - ln y))
  (D_x (fun _ : R => True) (ln y)) y (ln y)
y:R
Hy:0 < y
y <> 0
  assert (H0 := derivable_pt_lim_exp (ln y)); unfold derivable_pt_lim in H0;
    unfold limit1_in; unfold limit_in;
      simpl; unfold R_dist; intros; elim (H0 _ H);
        intros; exists (pos x); split.y:R
Hy:0 < y
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((exp (ln y + h) - exp (ln y)) / h -
     exp (ln y)) < eps0
eps:R
H:eps > 0
x:posreal
H1:forall h : R,
h <> 0 ->
Rabs h < x ->
Rabs
  ((exp (ln y + h) - exp (ln y)) / h - exp (ln y)) <
eps
x > 0
y:R
Hy:0 < y
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((exp (ln y + h) - exp (ln y)) / h -
     exp (ln y)) < eps0
eps:R
H:eps > 0
x:posreal
H1:forall h : R,
h <> 0 ->
Rabs h < x ->
Rabs
  ((exp (ln y + h) - exp (ln y)) / h - exp (ln y)) <
eps
forall x0 : R,
D_x (fun _ : R => True) (ln y) x0 /\
Rabs (x0 - ln y) < x ->
Rabs ((exp x0 - exp (ln y)) / (x0 - ln y) - y) < eps
y:R
Hy:0 < y
y <> 0
  apply (cond_pos x).y:R
Hy:0 < y
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((exp (ln y + h) - exp (ln y)) / h -
     exp (ln y)) < eps0
eps:R
H:eps > 0
x:posreal
H1:forall h : R,
h <> 0 ->
Rabs h < x ->
Rabs
  ((exp (ln y + h) - exp (ln y)) / h - exp (ln y)) <
eps
forall x0 : R,
D_x (fun _ : R => True) (ln y) x0 /\
Rabs (x0 - ln y) < x ->
Rabs ((exp x0 - exp (ln y)) / (x0 - ln y) - y) < eps
y:R
Hy:0 < y
y <> 0
  intros; pattern y at 3; rewrite <- exp_ln.y:R
Hy:0 < y
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((exp (ln y + h) - exp (ln y)) / h -
     exp (ln y)) < eps0
eps:R
H:eps > 0
x:posreal
H1:forall h : R,
h <> 0 ->
Rabs h < x ->
Rabs
  ((exp (ln y + h) - exp (ln y)) / h - exp (ln y)) <
eps
x0:R
H2:D_x (fun _ : R => True) (ln y) x0 /\
Rabs (x0 - ln y) < x
Rabs
  ((exp x0 - exp (ln y)) / (x0 - ln y) - exp (ln y)) <
eps
y:R
Hy:0 < y
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((exp (ln y + h) - exp (ln y)) / h -
     exp (ln y)) < eps0
eps:R
H:eps > 0
x:posreal
H1:forall h : R,
h <> 0 ->
Rabs h < x ->
Rabs
  ((exp (ln y + h) - exp (ln y)) / h - exp (ln y)) <
eps
x0:R
H2:D_x (fun _ : R => True) (ln y) x0 /\
Rabs (x0 - ln y) < x
0 < y
y:R
Hy:0 < y
y <> 0
  pattern x0 at 1; replace x0 with (ln y + (x0 - ln y));
    [ idtac | ring ].y:R
Hy:0 < y
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((exp (ln y + h) - exp (ln y)) / h -
     exp (ln y)) < eps0
eps:R
H:eps > 0
x:posreal
H1:forall h : R,
h <> 0 ->
Rabs h < x ->
Rabs
  ((exp (ln y + h) - exp (ln y)) / h - exp (ln y)) <
eps
x0:R
H2:D_x (fun _ : R => True) (ln y) x0 /\
Rabs (x0 - ln y) < x
Rabs
  ((exp (ln y + (x0 - ln y)) - exp (ln y)) /
   (x0 - ln y) - exp (ln y)) < eps
y:R
Hy:0 < y
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((exp (ln y + h) - exp (ln y)) / h -
     exp (ln y)) < eps0
eps:R
H:eps > 0
x:posreal
H1:forall h : R,
h <> 0 ->
Rabs h < x ->
Rabs
  ((exp (ln y + h) - exp (ln y)) / h - exp (ln y)) <
eps
x0:R
H2:D_x (fun _ : R => True) (ln y) x0 /\
Rabs (x0 - ln y) < x
0 < y
y:R
Hy:0 < y
y <> 0
  apply H1.y:R
Hy:0 < y
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((exp (ln y + h) - exp (ln y)) / h -
     exp (ln y)) < eps0
eps:R
H:eps > 0
x:posreal
H1:forall h : R,
h <> 0 ->
Rabs h < x ->
Rabs
  ((exp (ln y + h) - exp (ln y)) / h - exp (ln y)) <
eps
x0:R
H2:D_x (fun _ : R => True) (ln y) x0 /\
Rabs (x0 - ln y) < x
x0 - ln y <> 0
y:R
Hy:0 < y
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((exp (ln y + h) - exp (ln y)) / h -
     exp (ln y)) < eps0
eps:R
H:eps > 0
x:posreal
H1:forall h : R,
h <> 0 ->
Rabs h < x ->
Rabs
  ((exp (ln y + h) - exp (ln y)) / h - exp (ln y)) <
eps
x0:R
H2:D_x (fun _ : R => True) (ln y) x0 /\
Rabs (x0 - ln y) < x
Rabs (x0 - ln y) < x
y:R
Hy:0 < y
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((exp (ln y + h) - exp (ln y)) / h -
     exp (ln y)) < eps0
eps:R
H:eps > 0
x:posreal
H1:forall h : R,
h <> 0 ->
Rabs h < x ->
Rabs
  ((exp (ln y + h) - exp (ln y)) / h - exp (ln y)) <
eps
x0:R
H2:D_x (fun _ : R => True) (ln y) x0 /\
Rabs (x0 - ln y) < x
0 < y
y:R
Hy:0 < y
y <> 0
  elim H2; intros H3 _; unfold D_x in H3; elim H3; clear H3; intros _ H3;
    apply Rminus_eq_contra; apply (not_eq_sym (A:=R));
      apply H3.y:R
Hy:0 < y
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((exp (ln y + h) - exp (ln y)) / h -
     exp (ln y)) < eps0
eps:R
H:eps > 0
x:posreal
H1:forall h : R,
h <> 0 ->
Rabs h < x ->
Rabs
  ((exp (ln y + h) - exp (ln y)) / h - exp (ln y)) <
eps
x0:R
H2:D_x (fun _ : R => True) (ln y) x0 /\
Rabs (x0 - ln y) < x
Rabs (x0 - ln y) < x
y:R
Hy:0 < y
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((exp (ln y + h) - exp (ln y)) / h -
     exp (ln y)) < eps0
eps:R
H:eps > 0
x:posreal
H1:forall h : R,
h <> 0 ->
Rabs h < x ->
Rabs
  ((exp (ln y + h) - exp (ln y)) / h - exp (ln y)) <
eps
x0:R
H2:D_x (fun _ : R => True) (ln y) x0 /\
Rabs (x0 - ln y) < x
0 < y
y:R
Hy:0 < y
y <> 0
  elim H2; clear H2; intros _ H2; apply H2.y:R
Hy:0 < y
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((exp (ln y + h) - exp (ln y)) / h -
     exp (ln y)) < eps0
eps:R
H:eps > 0
x:posreal
H1:forall h : R,
h <> 0 ->
Rabs h < x ->
Rabs
  ((exp (ln y + h) - exp (ln y)) / h - exp (ln y)) <
eps
x0:R
H2:D_x (fun _ : R => True) (ln y) x0 /\
Rabs (x0 - ln y) < x
0 < y
y:R
Hy:0 < y
y <> 0
  assumption.y:R
Hy:0 < y
y <> 0
  red; intro; rewrite H in Hy; elim (Rlt_irrefl _ Hy).
Qed.

Lemma derivable_pt_lim_ln : forall x:R, 0 < x -> derivable_pt_lim ln x (/ x).forall x : R, 0 < x -> derivable_pt_lim ln x (/ x)
Proof.forall x : R, 0 < x -> derivable_pt_lim ln x (/ x)
  intros; assert (H0 := Dln x H); unfold D_in in H0; unfold limit1_in in H0;
    unfold limit_in in H0; simpl in H0; unfold R_dist in H0;
      unfold derivable_pt_lim; intros; elim (H0 _ H1);
        intros; elim H2; clear H2; intros; set (alp := Rmin x0 (x / 2));
          assert (H4 : 0 < alp).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
0 < alp
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((ln (x + h) - ln x) / h - / x) < eps
  unfold alp; unfold Rmin; case (Rle_dec x0 (x / 2)); intro.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
r:x0 <= x / 2
0 < x0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
n:~ x0 <= x / 2
0 < x / 2
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((ln (x + h) - ln x) / h - / x) < eps
  apply H2.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
n:~ x0 <= x / 2
0 < x / 2
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((ln (x + h) - ln x) / h - / x) < eps
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((ln (x + h) - ln x) / h - / x) < eps
  exists (mkposreal _ H4); intros; pattern h at 2;
    replace h with (x + h - x); [ idtac | ring ].x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs ((ln (x + h) - ln x) / (x + h - x) - / x) < eps
  apply H3; split.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
D_x (fun x1 : R => 0 < x1) x (x + h)
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  unfold D_x; split.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  destruct (Rcase_abs h) as [Hlt|Hgt].x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hgt:h >= 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  assert (H7 : Rabs h < x / 2).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
Rabs h < x / 2
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:Rabs h < x / 2
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hgt:h >= 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  apply Rlt_le_trans with alp.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
Rabs h < alp
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
alp <= x / 2
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:Rabs h < x / 2
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hgt:h >= 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  apply H6.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
alp <= x / 2
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:Rabs h < x / 2
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hgt:h >= 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  unfold alp; apply Rmin_r.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:Rabs h < x / 2
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hgt:h >= 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  apply Rlt_trans with (x / 2).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:Rabs h < x / 2
0 < x / 2
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:Rabs h < x / 2
x / 2 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hgt:h >= 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:Rabs h < x / 2
x / 2 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hgt:h >= 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  rewrite Rabs_left in H7.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:- h < x / 2
x / 2 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:Rabs h < x / 2
h < 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hgt:h >= 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  apply Rplus_lt_reg_l with (- h - x / 2).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:- h < x / 2
- h - x / 2 + x / 2 < - h - x / 2 + (x + h)
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:Rabs h < x / 2
h < 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hgt:h >= 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  replace (- h - x / 2 + x / 2) with (- h); [ idtac | ring ].x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:- h < x / 2
- h < - h - x / 2 + (x + h)
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:Rabs h < x / 2
h < 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hgt:h >= 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  pattern x at 2; rewrite double_var.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:- h < x / 2
- h < - h - x / 2 + (x / 2 + x / 2 + h)
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:Rabs h < x / 2
h < 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hgt:h >= 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  replace (- h - x / 2 + (x / 2 + x / 2 + h)) with (x / 2); [ apply H7 | ring ].x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hlt:h < 0
H7:Rabs h < x / 2
h < 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hgt:h >= 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  apply Hlt.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Hgt:h >= 0
0 < x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  apply Rplus_lt_le_0_compat; [ assumption | apply Rge_le; apply Hgt ].x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
x <> x + h
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  apply (sym_not_eq (A:=R)); apply Rminus_not_eq; replace (x + h - x) with h;
    [ apply H5 | ring ].x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x (fun x2 : R => 0 < x2) x x1 /\
   Rabs (x1 - x) < alp0 ->
   Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps0)
eps:R
H1:0 < eps
x0:R
H2:x0 > 0
H3:forall x1 : R,
D_x (fun x2 : R => 0 < x2) x x1 /\
Rabs (x1 - x) < x0 ->
Rabs ((ln x1 - ln x) / (x1 - x) - / x) < eps
alp:=Rmin x0 (x / 2):R
H4:0 < alp
h:R
H5:h <> 0
H6:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (x + h - x) < x0
  replace (x + h - x) with h;
  [ apply Rlt_le_trans with alp;
    [ apply H6 | unfold alp; apply Rmin_l ]
    | ring ].
Qed.

Theorem D_in_imp :
  forall (f g:R -> R) (D D1:R -> Prop) (x:R),
    (forall x:R, D1 x -> D x) -> D_in f g D x -> D_in f g D1 x.forall (f g : R -> R) (D D1 : R -> Prop) (x : R),
(forall x0 : R, D1 x0 -> D x0) ->
D_in f g D x -> D_in f g D1 x
Proof.forall (f g : R -> R) (D D1 : R -> Prop) (x : R),
(forall x0 : R, D1 x0 -> D x0) ->
D_in f g D x -> D_in f g D1 x
  intros f g D D1 x H; unfold D_in.f, g:R -> R
D, D1:R -> Prop
x:R
H:forall x0 : R, D1 x0 -> D x0
limit1_in (fun x0 : R => (f x0 - f x) / (x0 - x))
  (D_x D x) (g x) x ->
limit1_in (fun x0 : R => (f x0 - f x) / (x0 - x))
  (D_x D1 x) (g x) x
  intros H0; apply limit1_imp with (D := D_x D x); auto.f, g:R -> R
D, D1:R -> Prop
x:R
H:forall x0 : R, D1 x0 -> D x0
H0:limit1_in (fun x0 : R => (f x0 - f x) / (x0 - x))
  (D_x D x) (g x) x
forall x0 : R, D_x D1 x x0 -> D_x D x x0
  intros x1 [H1 H2]; split; auto.
Qed.

Theorem D_in_ext :
  forall (f g h:R -> R) (D:R -> Prop) (x:R),
    f x = g x -> D_in h f D x -> D_in h g D x.forall (f g h : R -> R) (D : R -> Prop) (x : R),
f x = g x -> D_in h f D x -> D_in h g D x
Proof.forall (f g h : R -> R) (D : R -> Prop) (x : R),
f x = g x -> D_in h f D x -> D_in h g D x
  intros f g h D x H; unfold D_in.f, g, h:R -> R
D:R -> Prop
x:R
H:f x = g x
limit1_in (fun x0 : R => (h x0 - h x) / (x0 - x))
  (D_x D x) (f x) x ->
limit1_in (fun x0 : R => (h x0 - h x) / (x0 - x))
  (D_x D x) (g x) x
  rewrite H; auto.
Qed.

Theorem Dpower :
  forall y z:R,
    0 < y ->
    D_in (fun x:R => x ^R z) (fun x:R => z * x ^R (z - 1)) (
      fun x:R => 0 < x) y.forall y z : R,
0 < y ->
D_in (fun x : R => x ^R z)
  (fun x : R => z * x ^R (z - 1)) (fun x : R => 0 < x)
  y
Proof.forall y z : R,
0 < y ->
D_in (fun x : R => x ^R z)
  (fun x : R => z * x ^R (z - 1)) (fun x : R => 0 < x)
  y
  intros y z H;
    apply D_in_imp with (D := Dgf (fun x:R => 0 < x) (fun x:R => True) ln).y, z:R
H:0 < y
forall x : R,
0 < x ->
Dgf (fun x0 : R => 0 < x0) (fun _ : R => True) ln x
y, z:R
H:0 < y
D_in (fun x : R => x ^R z)
  (fun x : R => z * x ^R (z - 1))
  (Dgf (fun x : R => 0 < x) (fun _ : R => True) ln) y
  intros x H0; repeat split.y, z:R
H:0 < y
x:R
H0:0 < x
0 < x
y, z:R
H:0 < y
D_in (fun x : R => x ^R z)
  (fun x : R => z * x ^R (z - 1))
  (Dgf (fun x : R => 0 < x) (fun _ : R => True) ln) y
  assumption.y, z:R
H:0 < y
D_in (fun x : R => x ^R z)
  (fun x : R => z * x ^R (z - 1))
  (Dgf (fun x : R => 0 < x) (fun _ : R => True) ln) y
  apply D_in_ext with (f := fun x:R => / x * (z * exp (z * ln x))).y, z:R
H:0 < y
/ y * (z * exp (z * ln y)) = z * y ^R (z - 1)
y, z:R
H:0 < y
D_in (fun x : R => x ^R z)
  (fun x : R => / x * (z * exp (z * ln x)))
  (Dgf (fun x : R => 0 < x) (fun _ : R => True) ln) y
  unfold Rminus; rewrite Rpower_plus; rewrite Rpower_Ropp;
    rewrite (Rpower_1 _ H); unfold Rpower; ring.y, z:R
H:0 < y
D_in (fun x : R => x ^R z)
  (fun x : R => / x * (z * exp (z * ln x)))
  (Dgf (fun x : R => 0 < x) (fun _ : R => True) ln) y
  apply Dcomp with
    (f := ln)
    (g := fun x:R => exp (z * x))
    (df := Rinv)
    (dg := fun x:R => z * exp (z * x)).y, z:R
H:0 < y
D_in ln Rinv (fun x : R => 0 < x) y
y, z:R
H:0 < y
D_in (fun x : R => exp (z * x))
  (fun x : R => z * exp (z * x)) (fun _ : R => True)
  (ln y)
  apply (Dln _ H).y, z:R
H:0 < y
D_in (fun x : R => exp (z * x))
  (fun x : R => z * exp (z * x)) (fun _ : R => True)
  (ln y)
  apply D_in_imp with
    (D := Dgf (fun x:R => True) (fun x:R => True) (fun x:R => z * x)).y, z:R
H:0 < y
forall x : R,
True ->
Dgf (fun _ : R => True) (fun _ : R => True)
  (fun x0 : R => z * x0) x
y, z:R
H:0 < y
D_in (fun x : R => exp (z * x))
  (fun x : R => z * exp (z * x))
  (Dgf (fun _ : R => True) (fun _ : R => True)
     (fun x : R => z * x)) (ln y)
  intros x H1; repeat split; auto.y, z:R
H:0 < y
D_in (fun x : R => exp (z * x))
  (fun x : R => z * exp (z * x))
  (Dgf (fun _ : R => True) (fun _ : R => True)
     (fun x : R => z * x)) (ln y)
  apply
    (Dcomp (fun _:R => True) (fun _:R => True) (fun x => z) exp
      (fun x:R => z * x) exp); simpl.y, z:R
H:0 < y
D_in (fun x : R => z * x) (fun _ : R => z)
  (fun _ : R => True) (ln y)
y, z:R
H:0 < y
D_in exp exp (fun _ : R => True) (z * ln y)
  apply D_in_ext with (f := fun x:R => z * 1).y, z:R
H:0 < y
z * 1 = z
y, z:R
H:0 < y
D_in (fun x : R => z * x) (fun _ : R => z * 1)
  (fun _ : R => True) (ln y)
y, z:R
H:0 < y
D_in exp exp (fun _ : R => True) (z * ln y)
  apply Rmult_1_r.y, z:R
H:0 < y
D_in (fun x : R => z * x) (fun _ : R => z * 1)
  (fun _ : R => True) (ln y)
y, z:R
H:0 < y
D_in exp exp (fun _ : R => True) (z * ln y)
  apply (Dmult_const (fun x => True) (fun x => x) (fun x => 1)); apply Dx.y, z:R
H:0 < y
D_in exp exp (fun _ : R => True) (z * ln y)
  assert (H0 := derivable_pt_lim_D_in exp exp (z * ln y)); elim H0; clear H0;
    intros _ H0; apply H0; apply derivable_pt_lim_exp.
Qed.

Theorem derivable_pt_lim_power :
  forall x y:R,
    0 < x -> derivable_pt_lim (fun x => x ^R y) x (y * x ^R (y - 1)).forall x y : R,
0 < x ->
derivable_pt_lim (fun x0 : R => x0 ^R y) x
  (y * x ^R (y - 1))
Proof.forall x y : R,
0 < x ->
derivable_pt_lim (fun x0 : R => x0 ^R y) x
  (y * x ^R (y - 1))
  intros x y H.x, y:R
H:0 < x
derivable_pt_lim (fun x0 : R => x0 ^R y) x
  (y * x ^R (y - 1))
  unfold Rminus; rewrite Rpower_plus.x, y:R
H:0 < x
derivable_pt_lim (fun x0 : R => x0 ^R y) x
  (y * (x ^R y * x ^R (- (1))))
  rewrite Rpower_Ropp.x, y:R
H:0 < x
derivable_pt_lim (fun x0 : R => x0 ^R y) x
  (y * (x ^R y * / x ^R 1))
  rewrite Rpower_1; auto.x, y:R
H:0 < x
derivable_pt_lim (fun x0 : R => x0 ^R y) x
  (y * (x ^R y * / x))
  rewrite <- Rmult_assoc.x, y:R
H:0 < x
derivable_pt_lim (fun x0 : R => x0 ^R y) x
  (y * x ^R y * / x)
  unfold Rpower.x, y:R
H:0 < x
derivable_pt_lim (fun x0 : R => exp (y * ln x0)) x
  (y * exp (y * ln x) * / x)
  apply derivable_pt_lim_comp with (f1 := ln) (f2 := fun x => exp (y * x)).x, y:R
H:0 < x
derivable_pt_lim ln x (/ x)
x, y:R
H:0 < x
derivable_pt_lim (fun x0 : R => exp (y * x0)) (ln x)
  (y * exp (y * ln x))
  apply derivable_pt_lim_ln; assumption.x, y:R
H:0 < x
derivable_pt_lim (fun x0 : R => exp (y * x0)) (ln x)
  (y * exp (y * ln x))
  rewrite (Rmult_comm y).x, y:R
H:0 < x
derivable_pt_lim (fun x0 : R => exp (y * x0)) (ln x)
  (exp (y * ln x) * y)
  apply derivable_pt_lim_comp with (f1 := fun x => y * x) (f2 := exp).x, y:R
H:0 < x
derivable_pt_lim (fun x0 : R => y * x0) (ln x) y
x, y:R
H:0 < x
derivable_pt_lim exp (y * ln x) (exp (y * ln x))
  pattern y at 2; replace y with (0 * ln x + y * 1).x, y:R
H:0 < x
derivable_pt_lim (fun x0 : R => y * x0) (ln x)
  (0 * ln x + y * 1)
x, y:R
H:0 < x
0 * ln x + y * 1 = y
x, y:R
H:0 < x
derivable_pt_lim exp (y * ln x) (exp (y * ln x))
  apply derivable_pt_lim_mult with (f1 := fun x:R => y) (f2 := fun x:R => x).x, y:R
H:0 < x
derivable_pt_lim (fun _ : R => y) (ln x) 0
x, y:R
H:0 < x
derivable_pt_lim (fun x0 : R => x0) (ln x) 1
x, y:R
H:0 < x
0 * ln x + y * 1 = y
x, y:R
H:0 < x
derivable_pt_lim exp (y * ln x) (exp (y * ln x))
  apply derivable_pt_lim_const with (a := y).x, y:R
H:0 < x
derivable_pt_lim (fun x0 : R => x0) (ln x) 1
x, y:R
H:0 < x
0 * ln x + y * 1 = y
x, y:R
H:0 < x
derivable_pt_lim exp (y * ln x) (exp (y * ln x))
  apply derivable_pt_lim_id.x, y:R
H:0 < x
0 * ln x + y * 1 = y
x, y:R
H:0 < x
derivable_pt_lim exp (y * ln x) (exp (y * ln x))
  ring.x, y:R
H:0 < x
derivable_pt_lim exp (y * ln x) (exp (y * ln x))
  apply derivable_pt_lim_exp.
Qed.

(* added later. *)

Lemma Rpower_mult_distr :  
  forall x y z, 0 < x -> 0 < y ->
   Rpower x z * Rpower y z = Rpower (x * y) z.forall x y z : R,
0 < x -> 0 < y -> x ^R z * y ^R z = (x * y) ^R z
intros x y z x0 y0; unfold Rpower.x, y, z:R
x0:0 < x
y0:0 < y
exp (z * ln x) * exp (z * ln y) = exp (z * ln (x * y))
rewrite <- exp_plus, ln_mult, Rmult_plus_distr_l; auto.
Qed.

Lemma Rlt_Rpower_l a b c: 0 < c -> 0 < a < b -> a ^R c < b ^R c.a, b, c:R
0 < c -> 0 < a < b -> a ^R c < b ^R c
Proof.a, b, c:R
0 < c -> 0 < a < b -> a ^R c < b ^R c
intros c0 [a0 ab]; apply exp_increasing.a, b, c:R
c0:0 < c
a0:0 < a
ab:a < b
c * ln a < c * ln b
now apply Rmult_lt_compat_l; auto; apply ln_increasing; lra.
Qed.

Lemma Rle_Rpower_l a b c: 0 <= c -> 0 < a <= b -> a ^R c <= b ^R c.a, b, c:R
0 <= c -> 0 < a <= b -> a ^R c <= b ^R c
Proof.a, b, c:R
0 <= c -> 0 < a <= b -> a ^R c <= b ^R c
intros [c0 | c0];
 [ | intros; rewrite <- c0, !Rpower_O; [apply Rle_refl | |] ].a, b, c:R
c0:0 < c
0 < a <= b -> a ^R c <= b ^R c
a, b, c:R
c0:0 = c
H:0 < a <= b
0 < b
a, b, c:R
c0:0 = c
H:0 < a <= b
0 < a
  intros [a0 [ab|ab]].a, b, c:R
c0:0 < c
a0:0 < a
ab:a < b
a ^R c <= b ^R c
a, b, c:R
c0:0 < c
a0:0 < a
ab:a = b
a ^R c <= b ^R c
a, b, c:R
c0:0 = c
H:0 < a <= b
0 < b
a, b, c:R
c0:0 = c
H:0 < a <= b
0 < a
  now apply Rlt_le, Rlt_Rpower_l;[ | split]; lra.a, b, c:R
c0:0 < c
a0:0 < a
ab:a = b
a ^R c <= b ^R c
a, b, c:R
c0:0 = c
H:0 < a <= b
0 < b
a, b, c:R
c0:0 = c
H:0 < a <= b
0 < a
  rewrite ab; apply Rle_refl.a, b, c:R
c0:0 = c
H:0 < a <= b
0 < b
a, b, c:R
c0:0 = c
H:0 < a <= b
0 < a
 apply Rlt_le_trans with a; tauto.a, b, c:R
c0:0 = c
H:0 < a <= b
0 < a
tauto.
Qed.

(* arcsinh function *)

Definition arcsinh x := ln (x + sqrt (x ^ 2 + 1)).

Lemma arcsinh_sinh : forall x, arcsinh (sinh x) = x.forall x : R, arcsinh (sinh x) = x
intros x; unfold sinh, arcsinh.x:R
ln
  ((exp x - exp (- x)) / 2 +
   sqrt (((exp x - exp (- x)) / 2) ^ 2 + 1)) = x
assert (Rminus_eq_0 : forall r, r - r = 0) by (intros; ring).x:R
Rminus_eq_0:forall r : R, r - r = 0
ln
  ((exp x - exp (- x)) / 2 +
   sqrt (((exp x - exp (- x)) / 2) ^ 2 + 1)) = x
rewrite <- exp_0, <- (Rminus_eq_0 x); unfold Rminus.x:R
Rminus_eq_0:forall r : R, r - r = 0
ln
  ((exp x + - exp (- x)) / 2 +
   sqrt
     (((exp x + - exp (- x)) / 2) ^ 2 + exp (x + - x))) =
x
rewrite exp_plus.x:R
Rminus_eq_0:forall r : R, r - r = 0
ln
  ((exp x + - exp (- x)) / 2 +
   sqrt
     (((exp x + - exp (- x)) / 2) ^ 2 +
      exp x * exp (- x))) = x
match goal with |- context[sqrt ?a] => 
  replace a with (((exp x + exp(-x))/2)^2) by field
end.x:R
Rminus_eq_0:forall r : R, r - r = 0
ln
  ((exp x + - exp (- x)) / 2 +
   sqrt (((exp x + exp (- x)) / 2) ^ 2)) = x
rewrite sqrt_pow2;
 [|apply Rlt_le, Rmult_lt_0_compat;[apply Rplus_lt_0_compat; apply exp_pos |
                            apply Rinv_0_lt_compat, Rlt_0_2]].x:R
Rminus_eq_0:forall r : R, r - r = 0
ln
  ((exp x + - exp (- x)) / 2 + (exp x + exp (- x)) / 2) =
x
match goal with |- context[ln ?a] => replace a with (exp x) by field end.x:R
Rminus_eq_0:forall r : R, r - r = 0
ln (exp x) = x 
rewrite ln_exp; reflexivity.
Qed.

Lemma sinh_arcsinh x : sinh (arcsinh x) = x.x:R
sinh (arcsinh x) = x 
unfold sinh, arcsinh.x:R
(exp (ln (x + sqrt (x ^ 2 + 1))) -
 exp (- ln (x + sqrt (x ^ 2 + 1)))) / 2 = x
assert (cmp : 0 < x + sqrt (x ^ 2 + 1)).x:R
0 < x + sqrt (x ^ 2 + 1)
x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
(exp (ln (x + sqrt (x ^ 2 + 1))) -
 exp (- ln (x + sqrt (x ^ 2 + 1)))) / 2 = x
 destruct (Rle_dec x 0).x:R
r:x <= 0
0 < x + sqrt (x ^ 2 + 1)
x:R
n:~ x <= 0
0 < x + sqrt (x ^ 2 + 1)
x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
(exp (ln (x + sqrt (x ^ 2 + 1))) -
 exp (- ln (x + sqrt (x ^ 2 + 1)))) / 2 = x
  replace (x ^ 2) with ((-x) ^ 2) by ring.x:R
r:x <= 0
0 < x + sqrt ((- x) ^ 2 + 1)
x:R
n:~ x <= 0
0 < x + sqrt (x ^ 2 + 1)
x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
(exp (ln (x + sqrt (x ^ 2 + 1))) -
 exp (- ln (x + sqrt (x ^ 2 + 1)))) / 2 = x
  assert (sqrt ((- x) ^ 2) < sqrt ((-x)^2+1)).x:R
r:x <= 0
sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
x:R
r:x <= 0
H:sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
0 < x + sqrt ((- x) ^ 2 + 1)
x:R
n:~ x <= 0
0 < x + sqrt (x ^ 2 + 1)
x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
(exp (ln (x + sqrt (x ^ 2 + 1))) -
 exp (- ln (x + sqrt (x ^ 2 + 1)))) / 2 = x
   apply sqrt_lt_1_alt.x:R
r:x <= 0
0 <= (- x) ^ 2 < (- x) ^ 2 + 1
x:R
r:x <= 0
H:sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
0 < x + sqrt ((- x) ^ 2 + 1)
x:R
n:~ x <= 0
0 < x + sqrt (x ^ 2 + 1)
x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
(exp (ln (x + sqrt (x ^ 2 + 1))) -
 exp (- ln (x + sqrt (x ^ 2 + 1)))) / 2 = x
   split;[apply pow_le | ]; lra.x:R
r:x <= 0
H:sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
0 < x + sqrt ((- x) ^ 2 + 1)
x:R
n:~ x <= 0
0 < x + sqrt (x ^ 2 + 1)
x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
(exp (ln (x + sqrt (x ^ 2 + 1))) -
 exp (- ln (x + sqrt (x ^ 2 + 1)))) / 2 = x
  pattern x at 1; replace x with (- (sqrt ((- x) ^ 2))).x:R
r:x <= 0
H:sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
0 < - sqrt ((- x) ^ 2) + sqrt ((- x) ^ 2 + 1)
x:R
r:x <= 0
H:sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
- sqrt ((- x) ^ 2) = x
x:R
n:~ x <= 0
0 < x + sqrt (x ^ 2 + 1)
x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
(exp (ln (x + sqrt (x ^ 2 + 1))) -
 exp (- ln (x + sqrt (x ^ 2 + 1)))) / 2 = x
   assert (t:= sqrt_pos ((-x)^2)); lra.x:R
r:x <= 0
H:sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
- sqrt ((- x) ^ 2) = x
x:R
n:~ x <= 0
0 < x + sqrt (x ^ 2 + 1)
x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
(exp (ln (x + sqrt (x ^ 2 + 1))) -
 exp (- ln (x + sqrt (x ^ 2 + 1)))) / 2 = x
  simpl; rewrite Rmult_1_r, sqrt_square, Ropp_involutive;[reflexivity | lra].x:R
n:~ x <= 0
0 < x + sqrt (x ^ 2 + 1)
x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
(exp (ln (x + sqrt (x ^ 2 + 1))) -
 exp (- ln (x + sqrt (x ^ 2 + 1)))) / 2 = x
  apply Rplus_lt_le_0_compat;[apply Rnot_le_gt; assumption | apply sqrt_pos].x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
(exp (ln (x + sqrt (x ^ 2 + 1))) -
 exp (- ln (x + sqrt (x ^ 2 + 1)))) / 2 = x
rewrite exp_ln;[ | assumption].x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
(x + sqrt (x ^ 2 + 1) -
 exp (- ln (x + sqrt (x ^ 2 + 1)))) / 2 = x
rewrite exp_Ropp, exp_ln;[ | assumption].x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
(x + sqrt (x ^ 2 + 1) - / (x + sqrt (x ^ 2 + 1))) / 2 =
x
assert (Rmult_minus_distr_r :
         forall x y z, (x - y) * z = x * z - y * z) by (intros; ring).x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
Rmult_minus_distr_r:forall x0 y z : R,
(x0 - y) * z = x0 * z - y * z
(x + sqrt (x ^ 2 + 1) - / (x + sqrt (x ^ 2 + 1))) / 2 =
x
apply Rminus_diag_uniq; unfold Rdiv; rewrite Rmult_minus_distr_r.x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
Rmult_minus_distr_r:forall x0 y z : R,
(x0 - y) * z = x0 * z - y * z
(x + sqrt (x ^ 2 + 1)) * / 2 -
/ (x + sqrt (x ^ 2 + 1)) * / 2 - x = 0
assert (t: forall x y z, x - z = y -> x - y - z = 0);[ | apply t; clear t].x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
Rmult_minus_distr_r:forall x0 y z : R,
(x0 - y) * z = x0 * z - y * z
forall x0 y z : R, x0 - z = y -> x0 - y - z = 0
x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
Rmult_minus_distr_r:forall x0 y z : R,
(x0 - y) * z = x0 * z - y * z
(x + sqrt (x ^ 2 + 1)) * / 2 - x =
/ (x + sqrt (x ^ 2 + 1)) * / 2
 intros a b c H; rewrite <- H; ring.x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
Rmult_minus_distr_r:forall x0 y z : R,
(x0 - y) * z = x0 * z - y * z
(x + sqrt (x ^ 2 + 1)) * / 2 - x =
/ (x + sqrt (x ^ 2 + 1)) * / 2
apply Rmult_eq_reg_l with (2 * (x + sqrt (x ^ 2 + 1)));[ |
 apply Rgt_not_eq, Rmult_lt_0_compat;[apply Rlt_0_2 | assumption]].x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
Rmult_minus_distr_r:forall x0 y z : R,
(x0 - y) * z = x0 * z - y * z
2 * (x + sqrt (x ^ 2 + 1)) *
((x + sqrt (x ^ 2 + 1)) * / 2 - x) =
2 * (x + sqrt (x ^ 2 + 1)) *
(/ (x + sqrt (x ^ 2 + 1)) * / 2)
assert (pow2_sqrt : forall x, 0 <= x -> sqrt x ^ 2 = x) by
 (intros; simpl; rewrite Rmult_1_r, sqrt_sqrt; auto).x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
Rmult_minus_distr_r:forall x0 y z : R,
(x0 - y) * z = x0 * z - y * z
pow2_sqrt:forall x0 : R, 0 <= x0 -> sqrt x0 ^ 2 = x0
2 * (x + sqrt (x ^ 2 + 1)) *
((x + sqrt (x ^ 2 + 1)) * / 2 - x) =
2 * (x + sqrt (x ^ 2 + 1)) *
(/ (x + sqrt (x ^ 2 + 1)) * / 2)
field_simplify;[rewrite pow2_sqrt;[field | ] | apply Rgt_not_eq; lra].x:R
cmp:0 < x + sqrt (x ^ 2 + 1)
Rmult_minus_distr_r:forall x0 y z : R,
(x0 - y) * z = x0 * z - y * z
pow2_sqrt:forall x0 : R, 0 <= x0 -> sqrt x0 ^ 2 = x0
0 <= x ^ 2 + 1
apply Rplus_le_le_0_compat;[simpl; rewrite Rmult_1_r; apply (Rle_0_sqr x)|apply Rlt_le, Rlt_0_1].
Qed.

Lemma derivable_pt_lim_arcsinh :
  forall x, derivable_pt_lim arcsinh x (/sqrt (x ^ 2 + 1)).forall x : R,
derivable_pt_lim arcsinh x (/ sqrt (x ^ 2 + 1))
intros x; unfold arcsinh.x:R
derivable_pt_lim
  (fun x0 : R => ln (x0 + sqrt (x0 ^ 2 + 1))) x
  (/ sqrt (x ^ 2 + 1))
assert (0 < x + sqrt (x ^ 2 + 1)).x:R
0 < x + sqrt (x ^ 2 + 1)
x:R
H:0 < x + sqrt (x ^ 2 + 1)
derivable_pt_lim
  (fun x0 : R => ln (x0 + sqrt (x0 ^ 2 + 1))) x
  (/ sqrt (x ^ 2 + 1))
 destruct (Rle_dec x 0);
  [ | assert (0 < x) by (apply Rnot_le_gt; assumption);
    apply Rplus_lt_le_0_compat; auto; apply sqrt_pos].x:R
r:x <= 0
0 < x + sqrt (x ^ 2 + 1)
x:R
H:0 < x + sqrt (x ^ 2 + 1)
derivable_pt_lim
  (fun x0 : R => ln (x0 + sqrt (x0 ^ 2 + 1))) x
  (/ sqrt (x ^ 2 + 1))
 replace (x ^ 2) with ((-x) ^ 2) by ring.x:R
r:x <= 0
0 < x + sqrt ((- x) ^ 2 + 1)
x:R
H:0 < x + sqrt (x ^ 2 + 1)
derivable_pt_lim
  (fun x0 : R => ln (x0 + sqrt (x0 ^ 2 + 1))) x
  (/ sqrt (x ^ 2 + 1))
 assert (sqrt ((- x) ^ 2) < sqrt ((-x)^2+1)).x:R
r:x <= 0
sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
x:R
r:x <= 0
H:sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
0 < x + sqrt ((- x) ^ 2 + 1)
x:R
H:0 < x + sqrt (x ^ 2 + 1)
derivable_pt_lim
  (fun x0 : R => ln (x0 + sqrt (x0 ^ 2 + 1))) x
  (/ sqrt (x ^ 2 + 1))
  apply sqrt_lt_1_alt.x:R
r:x <= 0
0 <= (- x) ^ 2 < (- x) ^ 2 + 1
x:R
r:x <= 0
H:sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
0 < x + sqrt ((- x) ^ 2 + 1)
x:R
H:0 < x + sqrt (x ^ 2 + 1)
derivable_pt_lim
  (fun x0 : R => ln (x0 + sqrt (x0 ^ 2 + 1))) x
  (/ sqrt (x ^ 2 + 1))
  split;[apply pow_le|]; lra.x:R
r:x <= 0
H:sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
0 < x + sqrt ((- x) ^ 2 + 1)
x:R
H:0 < x + sqrt (x ^ 2 + 1)
derivable_pt_lim
  (fun x0 : R => ln (x0 + sqrt (x0 ^ 2 + 1))) x
  (/ sqrt (x ^ 2 + 1))
 pattern x at 1; replace x with (- (sqrt ((- x) ^ 2))).x:R
r:x <= 0
H:sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
0 < - sqrt ((- x) ^ 2) + sqrt ((- x) ^ 2 + 1)
x:R
r:x <= 0
H:sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
- sqrt ((- x) ^ 2) = x
x:R
H:0 < x + sqrt (x ^ 2 + 1)
derivable_pt_lim
  (fun x0 : R => ln (x0 + sqrt (x0 ^ 2 + 1))) x
  (/ sqrt (x ^ 2 + 1))
  assert (t:= sqrt_pos ((-x)^2)); lra.x:R
r:x <= 0
H:sqrt ((- x) ^ 2) < sqrt ((- x) ^ 2 + 1)
- sqrt ((- x) ^ 2) = x
x:R
H:0 < x + sqrt (x ^ 2 + 1)
derivable_pt_lim
  (fun x0 : R => ln (x0 + sqrt (x0 ^ 2 + 1))) x
  (/ sqrt (x ^ 2 + 1))
 simpl; rewrite Rmult_1_r, sqrt_square, Ropp_involutive; auto; lra.x:R
H:0 < x + sqrt (x ^ 2 + 1)
derivable_pt_lim
  (fun x0 : R => ln (x0 + sqrt (x0 ^ 2 + 1))) x
  (/ sqrt (x ^ 2 + 1))
assert (0 < x ^ 2 + 1).x:R
H:0 < x + sqrt (x ^ 2 + 1)
0 < x ^ 2 + 1
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim
  (fun x0 : R => ln (x0 + sqrt (x0 ^ 2 + 1))) x
  (/ sqrt (x ^ 2 + 1))
 apply Rplus_le_lt_0_compat;[simpl; rewrite Rmult_1_r; apply Rle_0_sqr|lra].x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim
  (fun x0 : R => ln (x0 + sqrt (x0 ^ 2 + 1))) x
  (/ sqrt (x ^ 2 + 1))
replace (/sqrt (x ^ 2 + 1)) with
 (/(x + sqrt (x ^ 2 + 1)) * 
    (1 + (/(2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)))).x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim
  (fun x0 : R => ln (x0 + sqrt (x0 ^ 2 + 1))) x
  (/ (x + sqrt (x ^ 2 + 1)) *
   (1 + / (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
/ (x + sqrt (x ^ 2 + 1)) *
(1 + / (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)) =
/ sqrt (x ^ 2 + 1)
apply (derivable_pt_lim_comp (fun x => x + sqrt (x ^ 2 + 1)) ln).x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim
  (fun x0 : R => x0 + sqrt (x0 ^ 2 + 1)) x
  (1 + / (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim ln (x + sqrt (x ^ 2 + 1))
  (/ (x + sqrt (x ^ 2 + 1)))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
/ (x + sqrt (x ^ 2 + 1)) *
(1 + / (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)) =
/ sqrt (x ^ 2 + 1)
 apply (derivable_pt_lim_plus).x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim (fun x0 : R => x0) x 1
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim (fun x0 : R => sqrt (x0 ^ 2 + 1)) x
  (/ (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim ln (x + sqrt (x ^ 2 + 1))
  (/ (x + sqrt (x ^ 2 + 1)))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
/ (x + sqrt (x ^ 2 + 1)) *
(1 + / (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)) =
/ sqrt (x ^ 2 + 1)
  apply derivable_pt_lim_id.x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim (fun x0 : R => sqrt (x0 ^ 2 + 1)) x
  (/ (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim ln (x + sqrt (x ^ 2 + 1))
  (/ (x + sqrt (x ^ 2 + 1)))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
/ (x + sqrt (x ^ 2 + 1)) *
(1 + / (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)) =
/ sqrt (x ^ 2 + 1)
   apply (derivable_pt_lim_comp (fun x => x ^ 2 + 1) sqrt x).x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim (fun x0 : R => x0 ^ 2 + 1) x
  (INR 2 * x ^ 1 + 0)
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim sqrt (x ^ 2 + 1)
  (/ (2 * sqrt (x ^ 2 + 1)))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim ln (x + sqrt (x ^ 2 + 1))
  (/ (x + sqrt (x ^ 2 + 1)))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
/ (x + sqrt (x ^ 2 + 1)) *
(1 + / (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)) =
/ sqrt (x ^ 2 + 1)
    apply derivable_pt_lim_plus.x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim (fun x0 : R => x0 ^ 2) x
  (INR 2 * x ^ 1)
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim (fun _ : R => 1) x 0
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim sqrt (x ^ 2 + 1)
  (/ (2 * sqrt (x ^ 2 + 1)))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim ln (x + sqrt (x ^ 2 + 1))
  (/ (x + sqrt (x ^ 2 + 1)))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
/ (x + sqrt (x ^ 2 + 1)) *
(1 + / (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)) =
/ sqrt (x ^ 2 + 1)
     apply derivable_pt_lim_pow.x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim (fun _ : R => 1) x 0
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim sqrt (x ^ 2 + 1)
  (/ (2 * sqrt (x ^ 2 + 1)))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim ln (x + sqrt (x ^ 2 + 1))
  (/ (x + sqrt (x ^ 2 + 1)))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
/ (x + sqrt (x ^ 2 + 1)) *
(1 + / (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)) =
/ sqrt (x ^ 2 + 1)
    apply derivable_pt_lim_const.x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim sqrt (x ^ 2 + 1)
  (/ (2 * sqrt (x ^ 2 + 1)))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim ln (x + sqrt (x ^ 2 + 1))
  (/ (x + sqrt (x ^ 2 + 1)))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
/ (x + sqrt (x ^ 2 + 1)) *
(1 + / (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)) =
/ sqrt (x ^ 2 + 1)
   apply derivable_pt_lim_sqrt; assumption.x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
derivable_pt_lim ln (x + sqrt (x ^ 2 + 1))
  (/ (x + sqrt (x ^ 2 + 1)))
x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
/ (x + sqrt (x ^ 2 + 1)) *
(1 + / (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)) =
/ sqrt (x ^ 2 + 1)
  apply derivable_pt_lim_ln; assumption.x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
/ (x + sqrt (x ^ 2 + 1)) *
(1 + / (2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)) =
/ sqrt (x ^ 2 + 1)
 replace (INR 2 * x ^ 1 + 0) with (2 * x) by (simpl; ring).x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
/ (x + sqrt (x ^ 2 + 1)) *
(1 + / (2 * sqrt (x ^ 2 + 1)) * (2 * x)) =
/ sqrt (x ^ 2 + 1)
replace (1 + / (2 * sqrt (x ^ 2 + 1)) * (2 * x)) with
 (((sqrt (x ^ 2 + 1) + x))/sqrt (x ^ 2 + 1));
 [ | field; apply Rgt_not_eq, sqrt_lt_R0; assumption].x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
/ (x + sqrt (x ^ 2 + 1)) *
((sqrt (x ^ 2 + 1) + x) / sqrt (x ^ 2 + 1)) =
/ sqrt (x ^ 2 + 1)
apply Rmult_eq_reg_l with (x + sqrt (x ^ 2 + 1));
  [ | apply Rgt_not_eq; assumption].x:R
H:0 < x + sqrt (x ^ 2 + 1)
H0:0 < x ^ 2 + 1
(x + sqrt (x ^ 2 + 1)) *
(/ (x + sqrt (x ^ 2 + 1)) *
 ((sqrt (x ^ 2 + 1) + x) / sqrt (x ^ 2 + 1))) =
(x + sqrt (x ^ 2 + 1)) * / sqrt (x ^ 2 + 1)
rewrite <- Rmult_assoc, Rinv_r;[field | ]; apply Rgt_not_eq; auto;
  apply sqrt_lt_R0; assumption.
Qed.

Lemma arcsinh_lt : forall x y, x < y -> arcsinh x < arcsinh y.forall x y : R, x < y -> arcsinh x < arcsinh y
intros x y xy.x, y:R
xy:x < y
arcsinh x < arcsinh y
case (Rle_dec (arcsinh y) (arcsinh x));[ | apply Rnot_le_lt ].x, y:R
xy:x < y
arcsinh y <= arcsinh x -> arcsinh x < arcsinh y
intros abs; case (Rlt_not_le _ _ xy).x, y:R
xy:x < y
abs:arcsinh y <= arcsinh x
y <= x
rewrite <- (sinh_arcsinh y), <- (sinh_arcsinh x).x, y:R
xy:x < y
abs:arcsinh y <= arcsinh x
sinh (arcsinh y) <= sinh (arcsinh x)
destruct abs as [lt | q];[| rewrite q; lra].x, y:R
xy:x < y
lt:arcsinh y < arcsinh x
sinh (arcsinh y) <= sinh (arcsinh x)
apply Rlt_le, sinh_lt; assumption.
Qed.

Lemma arcsinh_le : forall x y, x <= y -> arcsinh x <= arcsinh y.forall x y : R, x <= y -> arcsinh x <= arcsinh y
intros x y [xy | xqy].x, y:R
xy:x < y
arcsinh x <= arcsinh y
x, y:R
xqy:x = y
arcsinh x <= arcsinh y
 apply Rlt_le, arcsinh_lt; assumption.x, y:R
xqy:x = y
arcsinh x <= arcsinh y
rewrite xqy; apply Rle_refl.
Qed.

Lemma arcsinh_0 : arcsinh 0 = 0.arcsinh 0 = 0
 unfold arcsinh; rewrite pow_ne_zero, !Rplus_0_l, sqrt_1, ln_1;
  [reflexivity | discriminate].
Qed.