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

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Export Rtrigo_fun.
Require Export Rtrigo_def.
Require Export Rtrigo_alt.
Require Export Cos_rel.
Require Export Cos_plus.
Require Import ZArith_base.
Require Import Zcomplements.
Require Import Lra.
Require Import Ranalysis1.
Require Import Rsqrt_def. 
Require Import PSeries_reg.

Local Open Scope nat_scope.
Local Open Scope R_scope.

Lemma CVN_R_cos :
  forall fn:nat -> R -> R,
    fn = (fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)) ->
    CVN_R fn.forall fn : nat -> R -> R,
fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)) ->
CVN_R fn
Proof.forall fn : nat -> R -> R,
fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)) ->
CVN_R fn
  unfold CVN_R in |- *; intros.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
CVN_r fn r
  cut ((r:R) <> 0).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0 -> CVN_r fn r
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  intro hyp_r; unfold CVN_r in |- *.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{An : nat -> R &
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun k : nat => Rabs (An k)) n) l /\
(forall (n : nat) (y : R),
 Boule 0 r y -> Rabs (fn n y) <= An n)}}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  exists (fun n:nat => / INR (fact (2 * n)) * r ^ (2 * n)).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l /\
(forall (n : nat) (y : R),
 Boule 0 r y ->
 Rabs (fn n y) <= / INR (fact (2 * n)) * r ^ (2 * n))}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  cut
    { l:R |
        Un_cv
        (fun n:nat =>
          sum_f_R0 (fun k:nat => Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
          n) l }.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l} ->
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l /\
(forall (n : nat) (y : R),
 Boule 0 r y ->
 Rabs (fn n y) <= / INR (fact (2 * n)) * r ^ (2 * n))}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  intros (x,p).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n)
  x
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l /\
(forall (n : nat) (y : R),
 Boule 0 r y ->
 Rabs (fn n y) <= / INR (fact (2 * n)) * r ^ (2 * n))}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  exists x.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n)
  x
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) x /\
(forall (n : nat) (y : R),
 Boule 0 r y ->
 Rabs (fn n y) <= / INR (fact (2 * n)) * r ^ (2 * n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  split.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n)
  x
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) x
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n)
  x
forall (n : nat) (y : R),
Boule 0 r y ->
Rabs (fn n y) <= / INR (fact (2 * n)) * r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply p.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n)
  x
forall (n : nat) (y : R),
Boule 0 r y ->
Rabs (fn n y) <= / INR (fact (2 * n)) * r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  intros; rewrite H; unfold Rdiv in |- *; do 2 rewrite Rabs_mult.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
Rabs ((-1) ^ n) * Rabs (/ INR (fact (2 * n))) *
Rabs (y ^ (2 * n)) <=
/ INR (fact (2 * n)) * r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite pow_1_abs; rewrite Rmult_1_l.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
Rabs (/ INR (fact (2 * n))) * Rabs (y ^ (2 * n)) <=
/ INR (fact (2 * n)) * r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  cut (0 < / INR (fact (2 * n))).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n)) ->
Rabs (/ INR (fact (2 * n))) * Rabs (y ^ (2 * n)) <=
/ INR (fact (2 * n)) * r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  intro; rewrite (Rabs_right _ (Rle_ge _ _ (Rlt_le _ _ H1))).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
H1:0 < / INR (fact (2 * n))
/ INR (fact (2 * n)) * Rabs (y ^ (2 * n)) <=
/ INR (fact (2 * n)) * r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rmult_le_compat_l.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
H1:0 < / INR (fact (2 * n))
0 <= / INR (fact (2 * n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
H1:0 < / INR (fact (2 * n))
Rabs (y ^ (2 * n)) <= r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  left; apply H1.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
H1:0 < / INR (fact (2 * n))
Rabs (y ^ (2 * n)) <= r ^ (2 * n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite <- RPow_abs; apply pow_maj_Rabs.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
H1:0 < / INR (fact (2 * n))
Rabs (Rabs y) <= r
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rabs_Rabsolu.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
H1:0 < / INR (fact (2 * n))
Rabs y <= r
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold Boule in H0; rewrite Rminus_0_r in H0.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Rabs y < r
H1:0 < / INR (fact (2 * n))
Rabs y <= r
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  left; apply H0.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x0 : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
     n0) x
n:nat
y:R
H0:Boule 0 r y
0 < / INR (fact (2 * n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rinv_0_lt_compat; apply INR_fact_lt_0.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) n) l}
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Alembert_C2.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
forall n : nat,
Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  intro; apply Rabs_no_R0.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
n:nat
/ INR (fact (2 * n)) * r ^ (2 * n) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply prod_neq_R0.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
n:nat
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
n:nat
r ^ (2 * n) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rinv_neq_0_compat.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
n:nat
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
n:nat
r ^ (2 * n) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply INR_fact_neq_0.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
n:nat
r ^ (2 * n) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply pow_nonzero; assumption.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  assert (H0 := Alembert_cos).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:Un_cv
  (fun n : nat => Rabs (cos_n (S n) / cos_n n)) 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold cos_n in H0; unfold Un_cv in H0; unfold Un_cv in |- *; intros.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
        Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) 0 <
  eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  cut (0 < eps / Rsqr r).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r² ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
        Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) 0 <
  eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  intro; elim (H0 _ H2); intros N0 H3.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n : nat,
(n >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n))))) 0 <
eps / r²
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
        Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) 0 <
  eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  exists N0; intros.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
R_dist
  (Rabs
     (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) 0 <
eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold R_dist in |- *; assert (H5 := H3 _ H4).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:R_dist
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n))))) 0 <
eps / r²
Rabs
  (Rabs
     (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
      Rabs (/ INR (fact (2 * n)) * r ^ (2 * n))) - 0) <
eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold R_dist in H5;
    replace
    (Rabs
      (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
        Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) with
    (Rsqr r *
      Rabs ((-1) ^ S n / INR (fact (2 * S n)) / ((-1) ^ n / INR (fact (2 * n))))).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs
  (r² *
   Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) < eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n)) /
   ((-1) ^ n / INR (fact (2 * n)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rmult_lt_reg_l with (/ Rsqr r).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
0 < / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ r² *
Rabs
  (r² *
   Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
/ r² * eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n)) /
   ((-1) ^ n / INR (fact (2 * n)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ r² *
Rabs
  (r² *
   Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
/ r² * eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n)) /
   ((-1) ^ n / INR (fact (2 * n)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  pattern (/ Rsqr r) at 1 in |- *; replace (/ Rsqr r) with (Rabs (/ Rsqr r)).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ r²) *
Rabs
  (r² *
   Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
/ r² * eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ r²) = / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n)) /
   ((-1) ^ n / INR (fact (2 * n)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite <- Rabs_mult; rewrite Rmult_minus_distr_l; rewrite Rmult_0_r;
    rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs
  (1 *
   Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
/ r² * eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ r²) = / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n)) /
   ((-1) ^ n / INR (fact (2 * n)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); apply H5.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ r²) = / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n)) /
   ((-1) ^ n / INR (fact (2 * n)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold Rsqr in |- *; apply prod_neq_R0; assumption.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ r²) = / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n)) /
   ((-1) ^ n / INR (fact (2 * n)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rabs_Rinv.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ Rabs r² = / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n)) /
   ((-1) ^ n / INR (fact (2 * n)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rabs_right.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ r² = / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n)) /
   ((-1) ^ n / INR (fact (2 * n)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  reflexivity.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n)) /
   ((-1) ^ n / INR (fact (2 * n)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rle_ge; apply Rle_0_sqr.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n)) /
   ((-1) ^ n / INR (fact (2 * n)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold Rsqr in |- *; apply prod_neq_R0; assumption.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² *
Rabs
  ((-1) ^ S n / INR (fact (2 * S n)) /
   ((-1) ^ n / INR (fact (2 * n)))) =
Rabs
  (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
   Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite (Rmult_comm (Rsqr r)); unfold Rdiv in |- *; repeat rewrite Rabs_mult;
    rewrite Rabs_Rabsolu; rewrite pow_1_abs; rewrite Rmult_1_l;
      repeat rewrite Rmult_assoc; apply Rmult_eq_compat_l.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ ((-1) ^ n * / INR (fact (2 * n)))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/
   (Rabs (/ INR (fact (2 * n))) * Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rabs_Rinv.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ Rabs ((-1) ^ n * / INR (fact (2 * n))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/
   (Rabs (/ INR (fact (2 * n))) * Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rabs_mult; rewrite (pow_1_abs n); rewrite Rmult_1_l;
    rewrite <- Rabs_Rinv.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ / INR (fact (2 * n))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/
   (Rabs (/ INR (fact (2 * n))) * Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rinv_involutive.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/
   (Rabs (/ INR (fact (2 * n))) * Rabs (r ^ (2 * n))))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rinv_mult_distr.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/ Rabs (/ INR (fact (2 * n))) *
   / Rabs (r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rabs_Rinv.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (/ / Rabs (INR (fact (2 * n))) *
   / Rabs (r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rinv_involutive.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) * r² =
Rabs (Rabs (r ^ (2 * S n))) *
Rabs
  (Rabs (INR (fact (2 * n))) * / Rabs (r ^ (2 * n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite (Rmult_comm (Rabs (Rabs (r ^ (2 * S n))))); rewrite Rabs_mult;
    rewrite Rabs_Rabsolu; rewrite Rmult_assoc; apply Rmult_eq_compat_l.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² =
Rabs (/ Rabs (r ^ (2 * n))) *
Rabs (Rabs (r ^ (2 * S n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  rewrite Rabs_Rinv.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² =
/ Rabs (Rabs (r ^ (2 * n))) *
Rabs (Rabs (r ^ (2 * S n)))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  do 2 rewrite Rabs_Rabsolu; repeat rewrite Rabs_right.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² = / r ^ (2 * n) * r ^ (2 * S n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  replace (r ^ (2 * S n)) with (r ^ (2 * n) * r * r).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² = / r ^ (2 * n) * (r ^ (2 * n) * r * r)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) * r * r = r ^ (2 * S n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r² = 1 * r * r
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) * r * r = r ^ (2 * S n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold Rsqr in |- *; ring.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) * r * r = r ^ (2 * S n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply pow_nonzero; assumption.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) * r * r = r ^ (2 * S n)
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  replace (2 * S n)%nat with (S (S (2 * n))).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) * r * r = r ^ S (S (2 * n))
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
S (S (2 * n)) = (2 * S n)%nat
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  simpl in |- *; ring.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
S (S (2 * n)) = (2 * S n)%nat
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  ring.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * S n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rle_ge; apply pow_le; left; apply (cond_pos r).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
r ^ (2 * n) >= 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rle_ge; apply pow_le; left; apply (cond_pos r).fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rabs_no_R0; apply pow_nonzero; assumption.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rabs_no_R0; apply INR_fact_neq_0.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply INR_fact_neq_0.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (/ INR (fact (2 * n))) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rabs_no_R0; apply Rinv_neq_0_compat; apply INR_fact_neq_0.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
Rabs (r ^ (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rabs_no_R0; apply pow_nonzero; assumption.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply INR_fact_neq_0.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rinv_neq_0_compat; apply INR_fact_neq_0.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n * / INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply prod_neq_R0.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
(-1) ^ n <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply pow_nonzero; discrR.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n0 / INR (fact (2 * S n0)) /
        ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
  eps0
eps:R
H1:eps > 0
H2:0 < eps / r²
N0:nat
H3:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs
     ((-1) ^ S n0 / INR (fact (2 * S n0)) /
      ((-1) ^ n0 / INR (fact (2 * n0))))) 0 <
eps / r²
n:nat
H4:(n >= N0)%nat
H5:Rabs
  (Rabs
     ((-1) ^ S n / INR (fact (2 * S n)) /
      ((-1) ^ n / INR (fact (2 * n)))) - 0) <
eps / r²
/ INR (fact (2 * n)) <> 0
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rinv_neq_0_compat; apply INR_fact_neq_0.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  unfold Rdiv in |- *; apply Rmult_lt_0_compat.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < eps
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply H1.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
hyp_r:(r : R) <> 0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       ((-1) ^ S n / INR (fact (2 * S n)) /
        ((-1) ^ n / INR (fact (2 * n))))) 0 <
  eps0
eps:R
H1:eps > 0
0 < / r²
fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption.fn:nat -> R -> R
H:fn =
(fun (N : nat) (x : R) =>
 (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N))
r:posreal
(r : R) <> 0
  assert (H0 := cond_pos r); red in |- *; intro; rewrite H1 in H0;
    elim (Rlt_irrefl _ H0).
Qed.

(**********)
Lemma continuity_cos : continuity cos.continuity cos
Proof.continuity cos
  set (fn := fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)).fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
continuity cos
  cut (CVN_R fn).fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn -> continuity cos
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  intro; cut (forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }).fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
(forall x : R,
 {l : R | Un_cv (fun N : nat => SP fn N x) l}) ->
continuity cos
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  intro cv; cut (forall n:nat, continuity (fn n)).fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
(forall n : nat, continuity (fn n)) -> continuity cos
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  intro; cut (forall x:R, cos x = SFL fn cv x).fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
(forall x : R, cos x = SFL fn cv x) -> continuity cos
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
forall x : R, cos x = SFL fn cv x
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  intro; cut (continuity (SFL fn cv) -> continuity cos).fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
H0:forall x : R, cos x = SFL fn cv x
(continuity (SFL fn cv) -> continuity cos) ->
continuity cos
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
H0:forall x : R, cos x = SFL fn cv x
continuity (SFL fn cv) -> continuity cos
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
forall x : R, cos x = SFL fn cv x
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  intro; apply H1.fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
H0:forall x : R, cos x = SFL fn cv x
H1:continuity (SFL fn cv) -> continuity cos
continuity (SFL fn cv)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
H0:forall x : R, cos x = SFL fn cv x
continuity (SFL fn cv) -> continuity cos
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
forall x : R, cos x = SFL fn cv x
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  apply SFL_continuity; assumption.fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
H0:forall x : R, cos x = SFL fn cv x
continuity (SFL fn cv) -> continuity cos
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
forall x : R, cos x = SFL fn cv x
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  unfold continuity in |- *; unfold continuity_pt in |- *;
    unfold continue_in in |- *; unfold limit1_in in |- *;
      unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *;
        intros.fn:=fun (N : nat) (x0 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H:forall n : nat, continuity (fn n)
H0:forall x0 : R, cos x0 = SFL fn cv x0
H1:forall x0 eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond x0 x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (SFL fn cv x1 - SFL fn cv x0) < eps0)
x, eps:R
H2:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (cos x0 - cos x) < eps)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
forall x : R, cos x = SFL fn cv x
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  elim (H1 x _ H2); intros.fn:=fun (N : nat) (x1 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x1 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H:forall n : nat, continuity (fn n)
H0:forall x1 : R, cos x1 = SFL fn cv x1
H1:forall x1 eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond x1 x2 /\ Rabs (x2 - x1) < alp ->
   Rabs (SFL fn cv x2 - SFL fn cv x1) < eps0)
x, eps:R
H2:eps > 0
x0:R
H3:x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (SFL fn cv x1 - SFL fn cv x) < eps)
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond x x1 /\ Rabs (x1 - x) < alp ->
   Rabs (cos x1 - cos x) < eps)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
forall x : R, cos x = SFL fn cv x
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  exists x0; intros.fn:=fun (N : nat) (x1 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x1 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H:forall n : nat, continuity (fn n)
H0:forall x1 : R, cos x1 = SFL fn cv x1
H1:forall x1 eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond x1 x2 /\ Rabs (x2 - x1) < alp ->
   Rabs (SFL fn cv x2 - SFL fn cv x1) < eps0)
x, eps:R
H2:eps > 0
x0:R
H3:x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (SFL fn cv x1 - SFL fn cv x) < eps)
x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (cos x1 - cos x) < eps)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
forall x : R, cos x = SFL fn cv x
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  elim H3; intros.fn:=fun (N : nat) (x1 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x1 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H:forall n : nat, continuity (fn n)
H0:forall x1 : R, cos x1 = SFL fn cv x1
H1:forall x1 eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond x1 x2 /\ Rabs (x2 - x1) < alp ->
   Rabs (SFL fn cv x2 - SFL fn cv x1) < eps0)
x, eps:R
H2:eps > 0
x0:R
H3:x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (SFL fn cv x1 - SFL fn cv x) < eps)
H4:x0 > 0
H5:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
Rabs (SFL fn cv x1 - SFL fn cv x) < eps
x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (cos x1 - cos x) < eps)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
forall x : R, cos x = SFL fn cv x
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  split.fn:=fun (N : nat) (x1 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x1 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H:forall n : nat, continuity (fn n)
H0:forall x1 : R, cos x1 = SFL fn cv x1
H1:forall x1 eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond x1 x2 /\ Rabs (x2 - x1) < alp ->
   Rabs (SFL fn cv x2 - SFL fn cv x1) < eps0)
x, eps:R
H2:eps > 0
x0:R
H3:x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (SFL fn cv x1 - SFL fn cv x) < eps)
H4:x0 > 0
H5:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
Rabs (SFL fn cv x1 - SFL fn cv x) < eps
x0 > 0
fn:=fun (N : nat) (x1 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x1 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H:forall n : nat, continuity (fn n)
H0:forall x1 : R, cos x1 = SFL fn cv x1
H1:forall x1 eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond x1 x2 /\ Rabs (x2 - x1) < alp ->
   Rabs (SFL fn cv x2 - SFL fn cv x1) < eps0)
x, eps:R
H2:eps > 0
x0:R
H3:x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (SFL fn cv x1 - SFL fn cv x) < eps)
H4:x0 > 0
H5:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
Rabs (SFL fn cv x1 - SFL fn cv x) < eps
forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
Rabs (cos x1 - cos x) < eps
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
forall x : R, cos x = SFL fn cv x
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  apply H4.fn:=fun (N : nat) (x1 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x1 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H:forall n : nat, continuity (fn n)
H0:forall x1 : R, cos x1 = SFL fn cv x1
H1:forall x1 eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x no_cond x1 x2 /\ Rabs (x2 - x1) < alp ->
   Rabs (SFL fn cv x2 - SFL fn cv x1) < eps0)
x, eps:R
H2:eps > 0
x0:R
H3:x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (SFL fn cv x1 - SFL fn cv x) < eps)
H4:x0 > 0
H5:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
Rabs (SFL fn cv x1 - SFL fn cv x) < eps
forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
Rabs (cos x1 - cos x) < eps
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
forall x : R, cos x = SFL fn cv x
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  intros; rewrite (H0 x); rewrite (H0 x1); apply H5; apply H6.fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H:forall n : nat, continuity (fn n)
forall x : R, cos x = SFL fn cv x
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  intro; unfold cos, SFL in |- *.fn:=fun (N : nat) (x0 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x0 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H:forall n : nat, continuity (fn n)
x:R
(let (a, _) := exist_cos x² in a) =
(let (a, _) := cv x in a)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  case (cv x) as (x1,HUn); case (exist_cos (Rsqr x)) as (x0,Hcos); intros.fn:=fun (N : nat) (x2 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x2 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x2 : R,
{l : R | Un_cv (fun N : nat => SP fn N x2) l}
H:forall n : nat, continuity (fn n)
x, x1:R
HUn:Un_cv (fun N : nat => SP fn N x) x1
x0:R
Hcos:cos_in x² x0
x0 = x1
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  symmetry; eapply UL_sequence.fn:=fun (N : nat) (x2 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x2 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x2 : R,
{l : R | Un_cv (fun N : nat => SP fn N x2) l}
H:forall n : nat, continuity (fn n)
x, x1:R
HUn:Un_cv (fun N : nat => SP fn N x) x1
x0:R
Hcos:cos_in x² x0
Un_cv ?Un x1
fn:=fun (N : nat) (x2 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x2 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x2 : R,
{l : R | Un_cv (fun N : nat => SP fn N x2) l}
H:forall n : nat, continuity (fn n)
x, x1:R
HUn:Un_cv (fun N : nat => SP fn N x) x1
x0:R
Hcos:cos_in x² x0
Un_cv ?Un x0
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  apply HUn.fn:=fun (N : nat) (x2 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x2 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x2 : R,
{l : R | Un_cv (fun N : nat => SP fn N x2) l}
H:forall n : nat, continuity (fn n)
x, x1:R
HUn:Un_cv (fun N : nat => SP fn N x) x1
x0:R
Hcos:cos_in x² x0
Un_cv (fun N : nat => SP fn N x) x0
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  unfold cos_in, infinite_sum in Hcos; unfold Un_cv in |- *; intros.fn:=fun (N : nat) (x2 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x2 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x2 : R,
{l : R | Un_cv (fun N : nat => SP fn N x2) l}
H:forall n : nat, continuity (fn n)
x, x1:R
HUn:Un_cv (fun N : nat => SP fn N x) x1
x0:R
Hcos:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => cos_n i * x² ^ i)
       n) x0 < eps0
eps:R
H0:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (SP fn n x) x0 < eps
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  elim (Hcos _ H0); intros N0 H1.fn:=fun (N : nat) (x2 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x2 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x2 : R,
{l : R | Un_cv (fun N : nat => SP fn N x2) l}
H:forall n : nat, continuity (fn n)
x, x1:R
HUn:Un_cv (fun N : nat => SP fn N x) x1
x0:R
Hcos:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => cos_n i * x² ^ i)
       n) x0 < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n : nat,
(n >= N0)%nat ->
R_dist
  (sum_f_R0 (fun i : nat => cos_n i * x² ^ i) n)
  x0 < eps
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (SP fn n x) x0 < eps
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  exists N0; intros.fn:=fun (N : nat) (x2 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x2 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x2 : R,
{l : R | Un_cv (fun N : nat => SP fn N x2) l}
H:forall n0 : nat, continuity (fn n0)
x, x1:R
HUn:Un_cv (fun N : nat => SP fn N x) x1
x0:R
Hcos:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => cos_n i * x² ^ i)
       n0) x0 < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0 (fun i : nat => cos_n i * x² ^ i) n0)
  x0 < eps
n:nat
H2:(n >= N0)%nat
R_dist (SP fn n x) x0 < eps
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  unfold R_dist in H1; unfold R_dist, SP in |- *.fn:=fun (N : nat) (x2 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x2 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x2 : R,
{l : R | Un_cv (fun N : nat => SP fn N x2) l}
H:forall n0 : nat, continuity (fn n0)
x, x1:R
HUn:Un_cv (fun N : nat => SP fn N x) x1
x0:R
Hcos:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => cos_n i * x² ^ i)
       n0) x0 < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * x² ^ i) n0 -
   x0) < eps
n:nat
H2:(n >= N0)%nat
Rabs (sum_f_R0 (fun k : nat => fn k x) n - x0) < eps
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  replace (sum_f_R0 (fun k:nat => fn k x) n) with
  (sum_f_R0 (fun i:nat => cos_n i * Rsqr x ^ i) n).fn:=fun (N : nat) (x2 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x2 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x2 : R,
{l : R | Un_cv (fun N : nat => SP fn N x2) l}
H:forall n0 : nat, continuity (fn n0)
x, x1:R
HUn:Un_cv (fun N : nat => SP fn N x) x1
x0:R
Hcos:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => cos_n i * x² ^ i)
       n0) x0 < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * x² ^ i) n0 -
   x0) < eps
n:nat
H2:(n >= N0)%nat
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * x² ^ i) n - x0) <
eps
fn:=fun (N : nat) (x2 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x2 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x2 : R,
{l : R | Un_cv (fun N : nat => SP fn N x2) l}
H:forall n0 : nat, continuity (fn n0)
x, x1:R
HUn:Un_cv (fun N : nat => SP fn N x) x1
x0:R
Hcos:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => cos_n i * x² ^ i)
       n0) x0 < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * x² ^ i) n0 -
   x0) < eps
n:nat
H2:(n >= N0)%nat
sum_f_R0 (fun i : nat => cos_n i * x² ^ i) n =
sum_f_R0 (fun k : nat => fn k x) n
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  apply H1; assumption.fn:=fun (N : nat) (x2 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x2 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x2 : R,
{l : R | Un_cv (fun N : nat => SP fn N x2) l}
H:forall n0 : nat, continuity (fn n0)
x, x1:R
HUn:Un_cv (fun N : nat => SP fn N x) x1
x0:R
Hcos:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0 (fun i : nat => cos_n i * x² ^ i)
       n0) x0 < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * x² ^ i) n0 -
   x0) < eps
n:nat
H2:(n >= N0)%nat
sum_f_R0 (fun i : nat => cos_n i * x² ^ i) n =
sum_f_R0 (fun k : nat => fn k x) n
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  apply sum_eq; intros.fn:=fun (N : nat) (x2 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x2 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x2 : R,
{l : R | Un_cv (fun N : nat => SP fn N x2) l}
H:forall n0 : nat, continuity (fn n0)
x, x1:R
HUn:Un_cv (fun N : nat => SP fn N x) x1
x0:R
Hcos:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i0 : nat => cos_n i0 * x² ^ i0) n0)
    x0 < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => cos_n i0 * x² ^ i0)
     n0 - x0) < eps
n:nat
H2:(n >= N0)%nat
i:nat
H3:(i <= n)%nat
cos_n i * x² ^ i = fn i x
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  unfold cos_n, fn in |- *; apply Rmult_eq_compat_l.fn:=fun (N : nat) (x2 : R) =>
(-1) ^ N / INR (fact (2 * N)) * x2 ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x2 : R,
{l : R | Un_cv (fun N : nat => SP fn N x2) l}
H:forall n0 : nat, continuity (fn n0)
x, x1:R
HUn:Un_cv (fun N : nat => SP fn N x) x1
x0:R
Hcos:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i0 : nat => cos_n i0 * x² ^ i0) n0)
    x0 < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => cos_n i0 * x² ^ i0)
     n0 - x0) < eps
n:nat
H2:(n >= N0)%nat
i:nat
H3:(i <= n)%nat
x² ^ i = x ^ (2 * i)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  unfold Rsqr in |- *; rewrite pow_sqr; reflexivity.fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  intro; unfold fn in |- *;
    replace (fun x:R => (-1) ^ n / INR (fact (2 * n)) * x ^ (2 * n)) with
    (fct_cte ((-1) ^ n / INR (fact (2 * n))) * pow_fct (2 * n))%F;
    [ idtac | reflexivity ].fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
n:nat
continuity
  (fct_cte ((-1) ^ n / INR (fact (2 * n))) *
   pow_fct (2 * n))
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  apply continuity_mult.fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
n:nat
continuity (fct_cte ((-1) ^ n / INR (fact (2 * n))))
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
n:nat
continuity (pow_fct (2 * n))
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  apply derivable_continuous; apply derivable_const.fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
n:nat
continuity (pow_fct (2 * n))
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  apply derivable_continuous; apply (derivable_pow (2 * n)).fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  apply CVN_R_CVS; apply X.fn:=fun (N : nat) (x : R) =>
(-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N):nat -> R -> R
CVN_R fn
  apply CVN_R_cos; unfold fn in |- *; reflexivity.
Qed.

Lemma sin_gt_cos_7_8 : sin (7 / 8) > cos (7 / 8).sin (7 / 8) > cos (7 / 8)
Proof.sin (7 / 8) > cos (7 / 8) 
assert (lo1 : 0 <= 7/8) by lra.lo1:0 <= 7 / 8
sin (7 / 8) > cos (7 / 8)
assert (up1 : 7/8 <= 4) by lra.lo1:0 <= 7 / 8
up1:7 / 8 <= 4
sin (7 / 8) > cos (7 / 8)
assert (lo : -2 <= 7/8) by lra.lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
sin (7 / 8) > cos (7 / 8)
assert (up : 7/8 <= 2) by lra.lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
up:7 / 8 <= 2
sin (7 / 8) > cos (7 / 8)
destruct (pre_sin_bound _ 0 lo1 up1) as [lower _ ].lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
up:7 / 8 <= 2
lower:sin_approx (7 / 8) (2 * 0 + 1) <= sin (7 / 8)
sin (7 / 8) > cos (7 / 8)
destruct (pre_cos_bound _ 0 lo up) as [_ upper].lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
up:7 / 8 <= 2
lower:sin_approx (7 / 8) (2 * 0 + 1) <= sin (7 / 8)
upper:cos (7 / 8) <=
cos_approx (7 / 8) (2 * (0 + 1))
sin (7 / 8) > cos (7 / 8)
apply Rle_lt_trans with (1 := upper).lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
up:7 / 8 <= 2
lower:sin_approx (7 / 8) (2 * 0 + 1) <= sin (7 / 8)
upper:cos (7 / 8) <=
cos_approx (7 / 8) (2 * (0 + 1))
cos_approx (7 / 8) (2 * (0 + 1)) < sin (7 / 8)
apply Rlt_le_trans with (2 := lower).lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
up:7 / 8 <= 2
lower:sin_approx (7 / 8) (2 * 0 + 1) <= sin (7 / 8)
upper:cos (7 / 8) <=
cos_approx (7 / 8) (2 * (0 + 1))
cos_approx (7 / 8) (2 * (0 + 1)) <
sin_approx (7 / 8) (2 * 0 + 1)
unfold cos_approx, sin_approx.lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
up:7 / 8 <= 2
lower:sin_approx (7 / 8) (2 * 0 + 1) <= sin (7 / 8)
upper:cos (7 / 8) <=
cos_approx (7 / 8) (2 * (0 + 1))
sum_f_R0 (cos_term (7 / 8)) (2 * (0 + 1)) <
sum_f_R0 (sin_term (7 / 8)) (2 * 0 + 1)
simpl sum_f_R0.lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
up:7 / 8 <= 2
lower:sin_approx (7 / 8) (2 * 0 + 1) <= sin (7 / 8)
upper:cos (7 / 8) <=
cos_approx (7 / 8) (2 * (0 + 1))
cos_term (7 / 8) 0 + cos_term (7 / 8) 1 +
cos_term (7 / 8) 2 <
sin_term (7 / 8) 0 + sin_term (7 / 8) 1
unfold cos_term, sin_term; simpl fact; rewrite !INR_IZR_INZ.lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
up:7 / 8 <= 2
lower:sin_approx (7 / 8) (2 * 0 + 1) <= sin (7 / 8)
upper:cos (7 / 8) <=
cos_approx (7 / 8) (2 * (0 + 1))
(-1) ^ 0 * ((7 / 8) ^ (2 * 0) / IZR (Z.of_nat 1)) +
(-1) ^ 1 * ((7 / 8) ^ (2 * 1) / IZR (Z.of_nat 2)) +
(-1) ^ 2 * ((7 / 8) ^ (2 * 2) / IZR (Z.of_nat 24)) <
(-1) ^ 0 * ((7 / 8) ^ (2 * 0 + 1) / IZR (Z.of_nat 1)) +
(-1) ^ 1 * ((7 / 8) ^ (2 * 1 + 1) / IZR (Z.of_nat 6))
simpl plus; simpl mult; simpl Z_of_nat.lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
up:7 / 8 <= 2
lower:sin_approx (7 / 8) (2 * 0 + 1) <= sin (7 / 8)
upper:cos (7 / 8) <=
cos_approx (7 / 8) (2 * (0 + 1))
(-1) ^ 0 * ((7 / 8) ^ 0 / 1) +
(-1) ^ 1 * ((7 / 8) ^ 2 / 2) +
(-1) ^ 2 * ((7 / 8) ^ 4 / 24) <
(-1) ^ 0 * ((7 / 8) ^ 1 / 1) +
(-1) ^ 1 * ((7 / 8) ^ 3 / 6)
field_simplify.lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
up:7 / 8 <= 2
lower:sin_approx (7 / 8) (2 * 0 + 1) <= sin (7 / 8)
upper:cos (7 / 8) <=
cos_approx (7 / 8) (2 * (0 + 1))
8073344 / 12582912 < 18760 / 24576
match goal with 
  |- IZR ?a / ?b < ?c / ?d =>
  apply Rmult_lt_reg_r with d;[apply (IZR_lt 0); reflexivity |
    unfold Rdiv at 2; rewrite Rmult_assoc, Rinv_l, Rmult_1_r, Rmult_comm;
     [ |apply not_eq_sym, Rlt_not_eq, (IZR_lt 0); reflexivity ]];
  apply Rmult_lt_reg_r with b;[apply (IZR_lt 0); reflexivity | ]
end.lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
up:7 / 8 <= 2
lower:sin_approx (7 / 8) (2 * 0 + 1) <= sin (7 / 8)
upper:cos (7 / 8) <=
cos_approx (7 / 8) (2 * (0 + 1))
24576 * (8073344 / 12582912) * 12582912 <
18760 * 12582912
unfold Rdiv; rewrite !Rmult_assoc, Rinv_l, Rmult_1_r;
 [ | apply not_eq_sym, Rlt_not_eq, (IZR_lt 0); reflexivity].lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
up:7 / 8 <= 2
lower:sin_approx (7 / 8) (2 * 0 + 1) <= sin (7 / 8)
upper:cos (7 / 8) <=
cos_approx (7 / 8) (2 * (0 + 1))
24576 * 8073344 < 18760 * 12582912
rewrite <- !mult_IZR.lo1:0 <= 7 / 8
up1:7 / 8 <= 4
lo:-2 <= 7 / 8
up:7 / 8 <= 2
lower:sin_approx (7 / 8) (2 * 0 + 1) <= sin (7 / 8)
upper:cos (7 / 8) <=
cos_approx (7 / 8) (2 * (0 + 1))
IZR (24576 * 8073344) < IZR (18760 * 12582912)
apply IZR_lt; reflexivity.
Qed.

Definition PI_2_aux : {z | 7/8 <= z <= 7/4 /\ -cos z = 0}.{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
assert (cc : continuity (fun r =>- cos r)).continuity (fun r : R => - cos r)
cc:continuity (fun r : R => - cos r)
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
 apply continuity_opp, continuity_cos.cc:continuity (fun r : R => - cos r)
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
assert (cvp : 0 < cos (7/8)).cc:continuity (fun r : R => - cos r)
0 < cos (7 / 8)
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
 assert (int78 : -2 <= 7/8 <= 2) by (split; lra).cc:continuity (fun r : R => - cos r)
int78:-2 <= 7 / 8 <= 2
0 < cos (7 / 8)
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
 destruct int78 as [lower upper].cc:continuity (fun r : R => - cos r)
lower:-2 <= 7 / 8
upper:7 / 8 <= 2
0 < cos (7 / 8)
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
 case (pre_cos_bound _ 0 lower upper).cc:continuity (fun r : R => - cos r)
lower:-2 <= 7 / 8
upper:7 / 8 <= 2
cos_approx (7 / 8) (2 * 0 + 1) <= cos (7 / 8) ->
cos (7 / 8) <= cos_approx (7 / 8) (2 * (0 + 1)) ->
0 < cos (7 / 8)
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
 unfold cos_approx; simpl sum_f_R0; unfold cos_term.cc:continuity (fun r : R => - cos r)
lower:-2 <= 7 / 8
upper:7 / 8 <= 2
(-1) ^ 0 * ((7 / 8) ^ (2 * 0) / INR (fact (2 * 0))) +
(-1) ^ 1 * ((7 / 8) ^ (2 * 1) / INR (fact (2 * 1))) <=
cos (7 / 8) ->
cos (7 / 8) <=
(-1) ^ 0 * ((7 / 8) ^ (2 * 0) / INR (fact (2 * 0))) +
(-1) ^ 1 * ((7 / 8) ^ (2 * 1) / INR (fact (2 * 1))) +
(-1) ^ 2 * ((7 / 8) ^ (2 * 2) / INR (fact (2 * 2))) ->
0 < cos (7 / 8)
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
 intros cl _; apply Rlt_le_trans with (2 := cl); simpl.cc:continuity (fun r : R => - cos r)
lower:-2 <= 7 / 8
upper:7 / 8 <= 2
cl:(-1) ^ 0 *
((7 / 8) ^ (2 * 0) / INR (fact (2 * 0))) +
(-1) ^ 1 *
((7 / 8) ^ (2 * 1) / INR (fact (2 * 1))) <=
cos (7 / 8)
0 <
1 * (1 / 1) + -1 * 1 * (7 / 8 * (7 / 8 * 1) / (1 + 1))
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
 lra.cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
assert (cun : cos (7/4) < 0).cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
cos (7 / 4) < 0
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
cun:cos (7 / 4) < 0
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
 replace (7/4) with (7/8 + 7/8) by field.cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
cos (7 / 8 + 7 / 8) < 0
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
cun:cos (7 / 4) < 0
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
 rewrite cos_plus.cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
cos (7 / 8) * cos (7 / 8) - sin (7 / 8) * sin (7 / 8) <
0
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
cun:cos (7 / 4) < 0
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
 apply Rlt_minus; apply Rsqr_incrst_1.cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
cos (7 / 8) < sin (7 / 8)
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
0 <= cos (7 / 8)
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
0 <= sin (7 / 8)
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
cun:cos (7 / 4) < 0
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
   exact sin_gt_cos_7_8.cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
0 <= cos (7 / 8)
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
0 <= sin (7 / 8)
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
cun:cos (7 / 4) < 0
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
  apply Rlt_le; assumption.cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
0 <= sin (7 / 8)
cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
cun:cos (7 / 4) < 0
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
 apply Rlt_le; apply Rlt_trans with (1 := cvp); exact sin_gt_cos_7_8.cc:continuity (fun r : R => - cos r)
cvp:0 < cos (7 / 8)
cun:cos (7 / 4) < 0
{z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}
apply IVT; auto; lra.
Qed.

Definition PI2 := proj1_sig PI_2_aux.

Definition PI := 2 * PI2.

Lemma cos_pi2 : cos PI2 = 0.cos PI2 = 0
unfold PI2; case PI_2_aux; simpl.forall x : R,
7 / 8 <= x <= 7 / 4 /\ - cos x = 0 -> cos x = 0
intros x [_ q]; rewrite <- (Ropp_involutive (cos x)), q; apply Ropp_0.
Qed.

Lemma pi2_int : 7/8 <= PI2 <= 7/4.7 / 8 <= PI2 <= 7 / 4
unfold PI2; case PI_2_aux; simpl; tauto.
Qed.

(**********)
Lemma cos_minus : forall x y:R, cos (x - y) = cos x * cos y + sin x * sin y.forall x y : R,
cos (x - y) = cos x * cos y + sin x * sin y
Proof.forall x y : R,
cos (x - y) = cos x * cos y + sin x * sin y
  intros; unfold Rminus in |- *; rewrite cos_plus.x, y:R
cos x * cos (- y) - sin x * sin (- y) =
cos x * cos y + sin x * sin y
  rewrite <- cos_sym; rewrite sin_antisym; ring.
Qed.

(**********)
Lemma sin2_cos2 : forall x:R, Rsqr (sin x) + Rsqr (cos x) = 1.forall x : R, (sin x)² + (cos x)² = 1
Proof.forall x : R, (sin x)² + (cos x)² = 1
  intro; unfold Rsqr in |- *; rewrite Rplus_comm; rewrite <- (cos_minus x x);
    unfold Rminus in |- *; rewrite Rplus_opp_r; apply cos_0.
Qed.

Lemma cos2 : forall x:R, Rsqr (cos x) = 1 - Rsqr (sin x).forall x : R, (cos x)² = 1 - (sin x)²
Proof.forall x : R, (cos x)² = 1 - (sin x)²
  intros x; rewrite <- (sin2_cos2 x); ring.
Qed.

Lemma sin2 : forall x:R, Rsqr (sin x) = 1 - Rsqr (cos x).forall x : R, (sin x)² = 1 - (cos x)²
Proof.forall x : R, (sin x)² = 1 - (cos x)²
  intro x; generalize (cos2 x); intro H1; rewrite H1.x:R
H1:(cos x)² = 1 - (sin x)²
(sin x)² = 1 - (1 - (sin x)²)
  unfold Rminus in |- *; rewrite Ropp_plus_distr; rewrite <- Rplus_assoc;
    rewrite Rplus_opp_r; rewrite Rplus_0_l; symmetry  in |- *;
      apply Ropp_involutive.
Qed.

(**********)
Lemma cos_PI2 : cos (PI / 2) = 0.cos (PI / 2) = 0
Proof.cos (PI / 2) = 0
 unfold PI; generalize cos_pi2; replace ((2 * PI2)/2) with PI2 by field; tauto.
Qed.

Lemma sin_pos_tech : forall x, 0 < x < 2 -> 0 < sin x.forall x : R, 0 < x < 2 -> 0 < sin x 
intros x [int1 int2].x:R
int1:0 < x
int2:x < 2
0 < sin x
assert (lo : 0 <= x) by (apply Rlt_le; assumption).x:R
int1:0 < x
int2:x < 2
lo:0 <= x
0 < sin x
assert (up : x <= 4) by (apply Rlt_le, Rlt_trans with (1:=int2); lra).x:R
int1:0 < x
int2:x < 2
lo:0 <= x
up:x <= 4
0 < sin x
destruct (pre_sin_bound _ 0 lo up) as [t _]; clear lo up.x:R
int1:0 < x
int2:x < 2
t:sin_approx x (2 * 0 + 1) <= sin x
0 < sin x
apply Rlt_le_trans with (2:= t); clear t.x:R
int1:0 < x
int2:x < 2
0 < sin_approx x (2 * 0 + 1)
unfold sin_approx; simpl sum_f_R0; unfold sin_term; simpl.x:R
int1:0 < x
int2:x < 2
0 <
1 * (x * 1 / 1) +
-1 * 1 * (x * (x * (x * 1)) / (1 + 1 + 1 + 1 + 1 + 1))
match goal with |- _ < ?a =>
  replace a with (x * (1 - x^2/6)) by (simpl; field)
end.x:R
int1:0 < x
int2:x < 2
0 < x * (1 - x ^ 2 / 6)
assert (t' : x ^ 2 <= 4).x:R
int1:0 < x
int2:x < 2
x ^ 2 <= 4
x:R
int1:0 < x
int2:x < 2
t':x ^ 2 <= 4
0 < x * (1 - x ^ 2 / 6)
 replace 4 with (2 ^ 2) by field.x:R
int1:0 < x
int2:x < 2
x ^ 2 <= 2 ^ 2
x:R
int1:0 < x
int2:x < 2
t':x ^ 2 <= 4
0 < x * (1 - x ^ 2 / 6)
 apply (pow_incr x 2); split; apply Rlt_le; assumption.x:R
int1:0 < x
int2:x < 2
t':x ^ 2 <= 4
0 < x * (1 - x ^ 2 / 6)
apply Rmult_lt_0_compat;[assumption | lra ].
Qed.

Lemma sin_PI2 : sin (PI / 2) = 1.sin (PI / 2) = 1
replace (PI / 2) with PI2 by (unfold PI; field).sin PI2 = 1
assert (int' : 0 < PI2 < 2).0 < PI2 < 2
int':0 < PI2 < 2
sin PI2 = 1
 destruct pi2_int; split; lra.int':0 < PI2 < 2
sin PI2 = 1
assert (lo2 := sin_pos_tech PI2 int').int':0 < PI2 < 2
lo2:0 < sin PI2
sin PI2 = 1
assert (t2 : Rabs (sin PI2) = 1).int':0 < PI2 < 2
lo2:0 < sin PI2
Rabs (sin PI2) = 1
int':0 < PI2 < 2
lo2:0 < sin PI2
t2:Rabs (sin PI2) = 1
sin PI2 = 1
 rewrite <- Rabs_R1; apply Rsqr_eq_abs_0.int':0 < PI2 < 2
lo2:0 < sin PI2
(sin PI2)² = 1²
int':0 < PI2 < 2
lo2:0 < sin PI2
t2:Rabs (sin PI2) = 1
sin PI2 = 1
 rewrite Rsqr_1, sin2, cos_pi2, Rsqr_0, Rminus_0_r; reflexivity.int':0 < PI2 < 2
lo2:0 < sin PI2
t2:Rabs (sin PI2) = 1
sin PI2 = 1
revert t2; rewrite Rabs_pos_eq;[| apply Rlt_le]; tauto.
Qed.

Lemma PI_RGT_0 : PI > 0.PI > 0
Proof.PI > 0 unfold PI; destruct pi2_int; lra. Qed.

Lemma PI_4 : PI <= 4.PI <= 4
Proof.PI <= 4 unfold PI; destruct pi2_int; lra. Qed.

(**********)
Lemma PI_neq0 : PI <> 0.PI <> 0
Proof.PI <> 0
  red in |- *; intro; assert (H0 := PI_RGT_0); rewrite H in H0;
    elim (Rlt_irrefl _ H0).
Qed.


(**********)
Lemma cos_PI : cos PI = -1.cos PI = -1
Proof.cos PI = -1
  replace PI with (PI / 2 + PI / 2).cos (PI / 2 + PI / 2) = -1
PI / 2 + PI / 2 = PI
  rewrite cos_plus.cos (PI / 2) * cos (PI / 2) -
sin (PI / 2) * sin (PI / 2) = -1
PI / 2 + PI / 2 = PI
  rewrite sin_PI2; rewrite cos_PI2.0 * 0 - 1 * 1 = -1
PI / 2 + PI / 2 = PI
  ring.PI / 2 + PI / 2 = PI
  symmetry  in |- *; apply double_var.
Qed.

Lemma sin_PI : sin PI = 0.sin PI = 0
Proof.sin PI = 0
  assert (H := sin2_cos2 PI).H:(sin PI)² + (cos PI)² = 1
sin PI = 0
  rewrite cos_PI in H.H:(sin PI)² + (-1)² = 1
sin PI = 0
  change (-1) with (-(1)) in H.H:(sin PI)² + (- (1))² = 1
sin PI = 0
  rewrite <- Rsqr_neg in H.H:(sin PI)² + 1² = 1
sin PI = 0
  rewrite Rsqr_1 in H.H:(sin PI)² + 1 = 1
sin PI = 0
  cut (Rsqr (sin PI) = 0).H:(sin PI)² + 1 = 1
(sin PI)² = 0 -> sin PI = 0
H:(sin PI)² + 1 = 1
(sin PI)² = 0
  intro; apply (Rsqr_eq_0 _ H0).H:(sin PI)² + 1 = 1
(sin PI)² = 0
  apply Rplus_eq_reg_l with 1.H:(sin PI)² + 1 = 1
1 + (sin PI)² = 1 + 0
  rewrite Rplus_0_r; rewrite Rplus_comm; exact H.
Qed.

Lemma sin_bound : forall (a : R) (n : nat), 0 <= a -> a <= PI ->
       sin_approx a (2 * n + 1) <= sin a <= sin_approx a (2 * (n + 1)).forall (a : R) (n : nat),
0 <= a ->
a <= PI ->
sin_approx a (2 * n + 1) <= sin a <=
sin_approx a (2 * (n + 1))
Proof.forall (a : R) (n : nat),
0 <= a ->
a <= PI ->
sin_approx a (2 * n + 1) <= sin a <=
sin_approx a (2 * (n + 1))
intros a n a0 api; apply pre_sin_bound.a:R
n:nat
a0:0 <= a
api:a <= PI
0 <= a
a:R
n:nat
a0:0 <= a
api:a <= PI
a <= 4
 assumption.a:R
n:nat
a0:0 <= a
api:a <= PI
a <= 4
apply Rle_trans with (1:= api) (2 := PI_4).
Qed.

Lemma cos_bound : forall (a : R) (n : nat), - PI / 2 <= a -> a <= PI / 2 ->
       cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1)).forall (a : R) (n : nat),
- PI / 2 <= a ->
a <= PI / 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
Proof.forall (a : R) (n : nat),
- PI / 2 <= a ->
a <= PI / 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
intros a n lower upper; apply pre_cos_bound.a:R
n:nat
lower:- PI / 2 <= a
upper:a <= PI / 2
-2 <= a
a:R
n:nat
lower:- PI / 2 <= a
upper:a <= PI / 2
a <= 2
 apply Rle_trans with (2 := lower).a:R
n:nat
lower:- PI / 2 <= a
upper:a <= PI / 2
-2 <= - PI / 2
a:R
n:nat
lower:- PI / 2 <= a
upper:a <= PI / 2
a <= 2
 apply Rmult_le_reg_r with 2; [lra |].a:R
n:nat
lower:- PI / 2 <= a
upper:a <= PI / 2
-2 * 2 <= - PI / 2 * 2
a:R
n:nat
lower:- PI / 2 <= a
upper:a <= PI / 2
a <= 2
 replace ((-PI/2) * 2) with (-PI) by field.a:R
n:nat
lower:- PI / 2 <= a
upper:a <= PI / 2
-2 * 2 <= - PI
a:R
n:nat
lower:- PI / 2 <= a
upper:a <= PI / 2
a <= 2
 assert (t := PI_4); lra.a:R
n:nat
lower:- PI / 2 <= a
upper:a <= PI / 2
a <= 2
apply Rle_trans with (1 := upper).a:R
n:nat
lower:- PI / 2 <= a
upper:a <= PI / 2
PI / 2 <= 2
apply Rmult_le_reg_r with 2; [lra | ].a:R
n:nat
lower:- PI / 2 <= a
upper:a <= PI / 2
PI / 2 * 2 <= 2 * 2
replace ((PI/2) * 2) with PI by field.a:R
n:nat
lower:- PI / 2 <= a
upper:a <= PI / 2
PI <= 2 * 2
generalize PI_4; intros; lra.
Qed.
(**********)
Lemma neg_cos : forall x:R, cos (x + PI) = - cos x.forall x : R, cos (x + PI) = - cos x
Proof.forall x : R, cos (x + PI) = - cos x
  intro x; rewrite cos_plus; rewrite sin_PI; rewrite cos_PI; ring.
Qed.

(**********)
Lemma sin_cos : forall x:R, sin x = - cos (PI / 2 + x).forall x : R, sin x = - cos (PI / 2 + x)
Proof.forall x : R, sin x = - cos (PI / 2 + x)
  intro x; rewrite cos_plus; rewrite sin_PI2; rewrite cos_PI2; ring.
Qed.

(**********)
Lemma sin_plus : forall x y:R, sin (x + y) = sin x * cos y + cos x * sin y.forall x y : R,
sin (x + y) = sin x * cos y + cos x * sin y
Proof.forall x y : R,
sin (x + y) = sin x * cos y + cos x * sin y
  intros.x, y:R
sin (x + y) = sin x * cos y + cos x * sin y
  rewrite (sin_cos (x + y)).x, y:R
- cos (PI / 2 + (x + y)) =
sin x * cos y + cos x * sin y
  replace (PI / 2 + (x + y)) with (PI / 2 + x + y); [ rewrite cos_plus | ring ].x, y:R
-
(cos (PI / 2 + x) * cos y - sin (PI / 2 + x) * sin y) =
sin x * cos y + cos x * sin y
  rewrite (sin_cos (PI / 2 + x)).x, y:R
-
(cos (PI / 2 + x) * cos y -
 - cos (PI / 2 + (PI / 2 + x)) * sin y) =
sin x * cos y + cos x * sin y
  replace (PI / 2 + (PI / 2 + x)) with (x + PI).x, y:R
- (cos (PI / 2 + x) * cos y - - cos (x + PI) * sin y) =
sin x * cos y + cos x * sin y
x, y:R
x + PI = PI / 2 + (PI / 2 + x)
  rewrite neg_cos.x, y:R
- (cos (PI / 2 + x) * cos y - - - cos x * sin y) =
sin x * cos y + cos x * sin y
x, y:R
x + PI = PI / 2 + (PI / 2 + x)
  replace (cos (PI / 2 + x)) with (- sin x).x, y:R
- (- sin x * cos y - - - cos x * sin y) =
sin x * cos y + cos x * sin y
x, y:R
- sin x = cos (PI / 2 + x)
x, y:R
x + PI = PI / 2 + (PI / 2 + x)
  ring.x, y:R
- sin x = cos (PI / 2 + x)
x, y:R
x + PI = PI / 2 + (PI / 2 + x)
  rewrite sin_cos; rewrite Ropp_involutive; reflexivity.x, y:R
x + PI = PI / 2 + (PI / 2 + x)
  pattern PI at 1 in |- *; rewrite (double_var PI); ring.
Qed.

Lemma sin_minus : forall x y:R, sin (x - y) = sin x * cos y - cos x * sin y.forall x y : R,
sin (x - y) = sin x * cos y - cos x * sin y
Proof.forall x y : R,
sin (x - y) = sin x * cos y - cos x * sin y
  intros; unfold Rminus in |- *; rewrite sin_plus.x, y:R
sin x * cos (- y) + cos x * sin (- y) =
sin x * cos y + - (cos x * sin y)
  rewrite <- cos_sym; rewrite sin_antisym; ring.
Qed.

(**********)
Definition tan (x:R) : R := sin x / cos x.

Lemma tan_plus :
  forall x y:R,
    cos x <> 0 ->
    cos y <> 0 ->
    cos (x + y) <> 0 ->
    1 - tan x * tan y <> 0 ->
    tan (x + y) = (tan x + tan y) / (1 - tan x * tan y).forall x y : R,
cos x <> 0 ->
cos y <> 0 ->
cos (x + y) <> 0 ->
1 - tan x * tan y <> 0 ->
tan (x + y) = (tan x + tan y) / (1 - tan x * tan y)
Proof.forall x y : R,
cos x <> 0 ->
cos y <> 0 ->
cos (x + y) <> 0 ->
1 - tan x * tan y <> 0 ->
tan (x + y) = (tan x + tan y) / (1 - tan x * tan y)
  intros; unfold tan in |- *; rewrite sin_plus; rewrite cos_plus;
    unfold Rdiv in |- *;
      replace (cos x * cos y - sin x * sin y) with
      (cos x * cos y * (1 - sin x * / cos x * (sin y * / cos y))).x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
(sin x * cos y + cos x * sin y) *
/
(cos x * cos y *
 (1 - sin x * / cos x * (sin y * / cos y))) =
(sin x * / cos x + sin y * / cos y) *
/ (1 - sin x * / cos x * (sin y * / cos y))
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y *
(1 - sin x * / cos x * (sin y * / cos y)) =
cos x * cos y - sin x * sin y
  rewrite Rinv_mult_distr.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
(sin x * cos y + cos x * sin y) *
(/ (cos x * cos y) *
 / (1 - sin x * / cos x * (sin y * / cos y))) =
(sin x * / cos x + sin y * / cos y) *
/ (1 - sin x * / cos x * (sin y * / cos y))
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
1 - sin x * / cos x * (sin y * / cos y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y *
(1 - sin x * / cos x * (sin y * / cos y)) =
cos x * cos y - sin x * sin y
  repeat rewrite <- Rmult_assoc;
    replace ((sin x * cos y + cos x * sin y) * / (cos x * cos y)) with
    (sin x * / cos x + sin y * / cos y).x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
(sin x * / cos x + sin y * / cos y) *
/ (1 - sin x * / cos x * sin y * / cos y) =
(sin x * / cos x + sin y * / cos y) *
/ (1 - sin x * / cos x * sin y * / cos y)
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
sin x * / cos x + sin y * / cos y =
(sin x * cos y + cos x * sin y) * / (cos x * cos y)
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
1 - sin x * / cos x * (sin y * / cos y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y *
(1 - sin x * / cos x * (sin y * / cos y)) =
cos x * cos y - sin x * sin y
  reflexivity.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
sin x * / cos x + sin y * / cos y =
(sin x * cos y + cos x * sin y) * / (cos x * cos y)
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
1 - sin x * / cos x * (sin y * / cos y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y *
(1 - sin x * / cos x * (sin y * / cos y)) =
cos x * cos y - sin x * sin y
  rewrite Rmult_plus_distr_r; rewrite Rinv_mult_distr.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
sin x * / cos x + sin y * / cos y =
sin x * cos y * (/ cos x * / cos y) +
cos x * sin y * (/ cos x * / cos y)
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
1 - sin x * / cos x * (sin y * / cos y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y *
(1 - sin x * / cos x * (sin y * / cos y)) =
cos x * cos y - sin x * sin y
  repeat rewrite Rmult_assoc; repeat rewrite (Rmult_comm (sin x));
    repeat rewrite <- Rmult_assoc.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
/ cos x * sin x + sin y * / cos y =
cos y * / cos x * / cos y * sin x +
cos x * sin y * / cos x * / cos y
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
1 - sin x * / cos x * (sin y * / cos y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y *
(1 - sin x * / cos x * (sin y * / cos y)) =
cos x * cos y - sin x * sin y
  repeat rewrite Rinv_r_simpl_m; [ reflexivity | assumption | assumption ].x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
1 - sin x * / cos x * (sin y * / cos y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y *
(1 - sin x * / cos x * (sin y * / cos y)) =
cos x * cos y - sin x * sin y
  assumption.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
1 - sin x * / cos x * (sin y * / cos y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y *
(1 - sin x * / cos x * (sin y * / cos y)) =
cos x * cos y - sin x * sin y
  assumption.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
1 - sin x * / cos x * (sin y * / cos y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y *
(1 - sin x * / cos x * (sin y * / cos y)) =
cos x * cos y - sin x * sin y
  apply prod_neq_R0; assumption.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
1 - sin x * / cos x * (sin y * / cos y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y *
(1 - sin x * / cos x * (sin y * / cos y)) =
cos x * cos y - sin x * sin y
  assumption.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x * cos y *
(1 - sin x * / cos x * (sin y * / cos y)) =
cos x * cos y - sin x * sin y
  unfold Rminus in |- *; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r;
    apply Rplus_eq_compat_l; repeat rewrite Rmult_assoc;
      rewrite (Rmult_comm (sin x)); rewrite (Rmult_comm (cos y));
        rewrite <- Ropp_mult_distr_r_reverse; repeat rewrite <- Rmult_assoc;
          rewrite <- Rinv_r_sym.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
1 * sin y * / cos y * - sin x * cos y =
- (sin x * sin y)
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x <> 0
  rewrite Rmult_1_l; rewrite (Rmult_comm (sin x));
    rewrite <- Ropp_mult_distr_r_reverse; repeat rewrite Rmult_assoc;
      apply Rmult_eq_compat_l; rewrite (Rmult_comm (/ cos y));
        rewrite Rmult_assoc; rewrite <- Rinv_r_sym.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
- sin x * 1 = - sin x
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x <> 0
  apply Rmult_1_r.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos y <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x <> 0
  assumption.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x + y) <> 0
H2:1 - tan x * tan y <> 0
cos x <> 0
  assumption.
Qed.

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

Some properties of cos, sin and tan

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

Lemma sin_2a : forall x:R, sin (2 * x) = 2 * sin x * cos x.forall x : R, sin (2 * x) = 2 * sin x * cos x
Proof.forall x : R, sin (2 * x) = 2 * sin x * cos x
  intro x; rewrite double; rewrite sin_plus.x:R
sin x * cos x + cos x * sin x = 2 * sin x * cos x
  rewrite <- (Rmult_comm (sin x)); symmetry  in |- *; rewrite Rmult_assoc;
    apply double.
Qed.

Lemma cos_2a : forall x:R, cos (2 * x) = cos x * cos x - sin x * sin x.forall x : R,
cos (2 * x) = cos x * cos x - sin x * sin x
Proof.forall x : R,
cos (2 * x) = cos x * cos x - sin x * sin x
  intro x; rewrite double; apply cos_plus.
Qed.

Lemma cos_2a_cos : forall x:R, cos (2 * x) = 2 * cos x * cos x - 1.forall x : R, cos (2 * x) = 2 * cos x * cos x - 1
Proof.forall x : R, cos (2 * x) = 2 * cos x * cos x - 1
  intro x; rewrite double; unfold Rminus in |- *; rewrite Rmult_assoc;
    rewrite cos_plus; generalize (sin2_cos2 x); rewrite double;
      intro H1; rewrite <- H1; ring_Rsqr.
Qed.

Lemma cos_2a_sin : forall x:R, cos (2 * x) = 1 - 2 * sin x * sin x.forall x : R, cos (2 * x) = 1 - 2 * sin x * sin x
Proof.forall x : R, cos (2 * x) = 1 - 2 * sin x * sin x
  intro x; rewrite Rmult_assoc; unfold Rminus in |- *; repeat rewrite double.x:R
cos (x + x) = 1 + - (sin x * sin x + sin x * sin x)
  generalize (sin2_cos2 x); intro H1; rewrite <- H1; rewrite cos_plus;
    ring_Rsqr.
Qed.

Lemma tan_2a :
  forall x:R,
    cos x <> 0 ->
    cos (2 * x) <> 0 ->
    1 - tan x * tan x <> 0 -> tan (2 * x) = 2 * tan x / (1 - tan x * tan x).forall x : R,
cos x <> 0 ->
cos (2 * x) <> 0 ->
1 - tan x * tan x <> 0 ->
tan (2 * x) = 2 * tan x / (1 - tan x * tan x)
Proof.forall x : R,
cos x <> 0 ->
cos (2 * x) <> 0 ->
1 - tan x * tan x <> 0 ->
tan (2 * x) = 2 * tan x / (1 - tan x * tan x)
  repeat rewrite double; intros; repeat rewrite double; rewrite double in H0;
    apply tan_plus; assumption.
Qed.

Lemma sin_neg : forall x:R, sin (- x) = - sin x.forall x : R, sin (- x) = - sin x
Proof.forall x : R, sin (- x) = - sin x
  apply sin_antisym.
Qed.

Lemma cos_neg : forall x:R, cos (- x) = cos x.forall x : R, cos (- x) = cos x
Proof.forall x : R, cos (- x) = cos x
  intro; symmetry  in |- *; apply cos_sym.
Qed.

Lemma tan_0 : tan 0 = 0.tan 0 = 0
Proof.tan 0 = 0
  unfold tan in |- *; rewrite sin_0; rewrite cos_0.0 / 1 = 0
  unfold Rdiv in |- *; apply Rmult_0_l.
Qed.

Lemma tan_neg : forall x:R, tan (- x) = - tan x.forall x : R, tan (- x) = - tan x
Proof.forall x : R, tan (- x) = - tan x
  intros x; unfold tan in |- *; rewrite sin_neg; rewrite cos_neg;
    unfold Rdiv in |- *.x:R
- sin x * / cos x = - (sin x * / cos x)
  apply Ropp_mult_distr_l_reverse.
Qed.

Lemma tan_minus :
  forall x y:R,
    cos x <> 0 ->
    cos y <> 0 ->
    cos (x - y) <> 0 ->
    1 + tan x * tan y <> 0 ->
    tan (x - y) = (tan x - tan y) / (1 + tan x * tan y).forall x y : R,
cos x <> 0 ->
cos y <> 0 ->
cos (x - y) <> 0 ->
1 + tan x * tan y <> 0 ->
tan (x - y) = (tan x - tan y) / (1 + tan x * tan y)
Proof.forall x y : R,
cos x <> 0 ->
cos y <> 0 ->
cos (x - y) <> 0 ->
1 + tan x * tan y <> 0 ->
tan (x - y) = (tan x - tan y) / (1 + tan x * tan y)
  intros; unfold Rminus in |- *; rewrite tan_plus.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
(tan x + tan (- y)) / (1 - tan x * tan (- y)) =
(tan x + - tan y) / (1 + tan x * tan y)
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
cos x <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
cos (- y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
cos (x + - y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
1 - tan x * tan (- y) <> 0
  rewrite tan_neg; unfold Rminus in |- *; rewrite <- Ropp_mult_distr_l_reverse;
    rewrite Rmult_opp_opp; reflexivity.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
cos x <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
cos (- y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
cos (x + - y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
1 - tan x * tan (- y) <> 0
  assumption.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
cos (- y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
cos (x + - y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
1 - tan x * tan (- y) <> 0
  rewrite cos_neg; assumption.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
cos (x + - y) <> 0
x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
1 - tan x * tan (- y) <> 0
  assumption.x, y:R
H:cos x <> 0
H0:cos y <> 0
H1:cos (x - y) <> 0
H2:1 + tan x * tan y <> 0
1 - tan x * tan (- y) <> 0
  rewrite tan_neg; unfold Rminus in |- *; rewrite <- Ropp_mult_distr_l_reverse;
    rewrite Rmult_opp_opp; assumption.
Qed.

Lemma cos_3PI2 : cos (3 * (PI / 2)) = 0.cos (3 * (PI / 2)) = 0
Proof.cos (3 * (PI / 2)) = 0
  replace (3 * (PI / 2)) with (PI + PI / 2).cos (PI + PI / 2) = 0
PI + PI / 2 = 3 * (PI / 2)
  rewrite cos_plus; rewrite sin_PI; rewrite cos_PI2; ring.PI + PI / 2 = 3 * (PI / 2)
  pattern PI at 1 in |- *; rewrite (double_var PI).PI / 2 + PI / 2 + PI / 2 = 3 * (PI / 2)
  ring.
Qed.

Lemma sin_2PI : sin (2 * PI) = 0.sin (2 * PI) = 0
Proof.sin (2 * PI) = 0
  rewrite sin_2a; rewrite sin_PI; ring.
Qed.

Lemma cos_2PI : cos (2 * PI) = 1.cos (2 * PI) = 1
Proof.cos (2 * PI) = 1
  rewrite cos_2a; rewrite sin_PI; rewrite cos_PI; ring.
Qed.

Lemma neg_sin : forall x:R, sin (x + PI) = - sin x.forall x : R, sin (x + PI) = - sin x
Proof.forall x : R, sin (x + PI) = - sin x
  intro x; rewrite sin_plus; rewrite sin_PI; rewrite cos_PI; ring.
Qed.

Lemma sin_PI_x : forall x:R, sin (PI - x) = sin x.forall x : R, sin (PI - x) = sin x
Proof.forall x : R, sin (PI - x) = sin x
  intro x; rewrite sin_minus; rewrite sin_PI; rewrite cos_PI.x:R
0 * cos x - -1 * sin x = sin x
  ring.
Qed.

Lemma sin_period : forall (x:R) (k:nat), sin (x + 2 * INR k * PI) = sin x.forall (x : R) (k : nat),
sin (x + 2 * INR k * PI) = sin x
Proof.forall (x : R) (k : nat),
sin (x + 2 * INR k * PI) = sin x
  intros x k; induction  k as [| k Hreck].x:R
sin (x + 2 * INR 0 * PI) = sin x
x:R
k:nat
Hreck:sin (x + 2 * INR k * PI) = sin x
sin (x + 2 * INR (S k) * PI) = sin x
  simpl in |- *;  ring_simplify (x + 2 * 0 * PI).x:R
sin x = sin x
x:R
k:nat
Hreck:sin (x + 2 * INR k * PI) = sin x
sin (x + 2 * INR (S k) * PI) = sin x
  trivial.x:R
k:nat
Hreck:sin (x + 2 * INR k * PI) = sin x
sin (x + 2 * INR (S k) * PI) = sin x

  replace (x + 2 * INR (S k) * PI) with (x + 2 * INR k * PI + 2 * PI).x:R
k:nat
Hreck:sin (x + 2 * INR k * PI) = sin x
sin (x + 2 * INR k * PI + 2 * PI) = sin x
x:R
k:nat
Hreck:sin (x + 2 * INR k * PI) = sin x
x + 2 * INR k * PI + 2 * PI = x + 2 * INR (S k) * PI
  rewrite sin_plus in |- *; rewrite sin_2PI in |- *; rewrite cos_2PI in |- *.x:R
k:nat
Hreck:sin (x + 2 * INR k * PI) = sin x
sin (x + 2 * INR k * PI) * 1 +
cos (x + 2 * INR k * PI) * 0 = 
sin x
x:R
k:nat
Hreck:sin (x + 2 * INR k * PI) = sin x
x + 2 * INR k * PI + 2 * PI = x + 2 * INR (S k) * PI
  ring_simplify; trivial.x:R
k:nat
Hreck:sin (x + 2 * INR k * PI) = sin x
x + 2 * INR k * PI + 2 * PI = x + 2 * INR (S k) * PI
  rewrite S_INR in |- *;  ring.
Qed.

Lemma cos_period : forall (x:R) (k:nat), cos (x + 2 * INR k * PI) = cos x.forall (x : R) (k : nat),
cos (x + 2 * INR k * PI) = cos x
Proof.forall (x : R) (k : nat),
cos (x + 2 * INR k * PI) = cos x
  intros x k; induction  k as [| k Hreck].x:R
cos (x + 2 * INR 0 * PI) = cos x
x:R
k:nat
Hreck:cos (x + 2 * INR k * PI) = cos x
cos (x + 2 * INR (S k) * PI) = cos x
  simpl in |- *;  ring_simplify (x + 2 * 0 * PI).x:R
cos x = cos x
x:R
k:nat
Hreck:cos (x + 2 * INR k * PI) = cos x
cos (x + 2 * INR (S k) * PI) = cos x
  trivial.x:R
k:nat
Hreck:cos (x + 2 * INR k * PI) = cos x
cos (x + 2 * INR (S k) * PI) = cos x

  replace (x + 2 * INR (S k) * PI) with (x + 2 * INR k * PI + 2 * PI).x:R
k:nat
Hreck:cos (x + 2 * INR k * PI) = cos x
cos (x + 2 * INR k * PI + 2 * PI) = cos x
x:R
k:nat
Hreck:cos (x + 2 * INR k * PI) = cos x
x + 2 * INR k * PI + 2 * PI = x + 2 * INR (S k) * PI
  rewrite cos_plus in |- *; rewrite sin_2PI in |- *; rewrite cos_2PI in |- *.x:R
k:nat
Hreck:cos (x + 2 * INR k * PI) = cos x
cos (x + 2 * INR k * PI) * 1 -
sin (x + 2 * INR k * PI) * 0 = 
cos x
x:R
k:nat
Hreck:cos (x + 2 * INR k * PI) = cos x
x + 2 * INR k * PI + 2 * PI = x + 2 * INR (S k) * PI
  ring_simplify; trivial.x:R
k:nat
Hreck:cos (x + 2 * INR k * PI) = cos x
x + 2 * INR k * PI + 2 * PI = x + 2 * INR (S k) * PI
  rewrite S_INR in |- *;  ring.
Qed.

Lemma sin_shift : forall x:R, sin (PI / 2 - x) = cos x.forall x : R, sin (PI / 2 - x) = cos x
Proof.forall x : R, sin (PI / 2 - x) = cos x
  intro x; rewrite sin_minus; rewrite sin_PI2; rewrite cos_PI2; ring.
Qed.

Lemma cos_shift : forall x:R, cos (PI / 2 - x) = sin x.forall x : R, cos (PI / 2 - x) = sin x
Proof.forall x : R, cos (PI / 2 - x) = sin x
  intro x; rewrite cos_minus; rewrite sin_PI2; rewrite cos_PI2; ring.
Qed.

Lemma cos_sin : forall x:R, cos x = sin (PI / 2 + x).forall x : R, cos x = sin (PI / 2 + x)
Proof.forall x : R, cos x = sin (PI / 2 + x)
  intro x; rewrite sin_plus; rewrite sin_PI2; rewrite cos_PI2; ring.
Qed.

Lemma PI2_RGT_0 : 0 < PI / 2.0 < PI / 2
Proof.0 < PI / 2
  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
    [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup ].
Qed.

Lemma SIN_bound : forall x:R, -1 <= sin x <= 1.forall x : R, -1 <= sin x <= 1
Proof.forall x : R, -1 <= sin x <= 1
  intro; destruct (Rle_dec (-1) (sin x)) as [Hle|Hnle].x:R
Hle:-1 <= sin x
-1 <= sin x <= 1
x:R
Hnle:~ -1 <= sin x
-1 <= sin x <= 1
  destruct (Rle_dec (sin x) 1) as [Hle'|Hnle'].x:R
Hle:-1 <= sin x
Hle':sin x <= 1
-1 <= sin x <= 1
x:R
Hle:-1 <= sin x
Hnle':~ sin x <= 1
-1 <= sin x <= 1
x:R
Hnle:~ -1 <= sin x
-1 <= sin x <= 1
  split; assumption.x:R
Hle:-1 <= sin x
Hnle':~ sin x <= 1
-1 <= sin x <= 1
x:R
Hnle:~ -1 <= sin x
-1 <= sin x <= 1
  cut (1 < sin x).x:R
Hle:-1 <= sin x
Hnle':~ sin x <= 1
1 < sin x -> -1 <= sin x <= 1
x:R
Hle:-1 <= sin x
Hnle':~ sin x <= 1
1 < sin x
x:R
Hnle:~ -1 <= sin x
-1 <= sin x <= 1
  intro;
    generalize
      (Rsqr_incrst_1 1 (sin x) H (Rlt_le 0 1 Rlt_0_1)
        (Rlt_le 0 (sin x) (Rlt_trans 0 1 (sin x) Rlt_0_1 H)));
      rewrite Rsqr_1; intro; rewrite sin2 in H0; unfold Rminus in H0.x:R
Hle:-1 <= sin x
Hnle':~ sin x <= 1
H:1 < sin x
H0:1 < 1 + - (cos x)²
-1 <= sin x <= 1
x:R
Hle:-1 <= sin x
Hnle':~ sin x <= 1
1 < sin x
x:R
Hnle:~ -1 <= sin x
-1 <= sin x <= 1
        generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0);
          repeat rewrite <- Rplus_assoc; change (-1) with (-(1)); rewrite Rplus_opp_l;
            rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1;
              generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1);
                repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x));
                  intro; elim (Rlt_irrefl 0 (Rle_lt_trans 0 (Rsqr (cos x)) 0 H3 H2)).x:R
Hle:-1 <= sin x
Hnle':~ sin x <= 1
1 < sin x
x:R
Hnle:~ -1 <= sin x
-1 <= sin x <= 1
  auto with real.x:R
Hnle:~ -1 <= sin x
-1 <= sin x <= 1
  cut (sin x < -1).x:R
Hnle:~ -1 <= sin x
sin x < -1 -> -1 <= sin x <= 1
x:R
Hnle:~ -1 <= sin x
sin x < -1
  intro; generalize (Ropp_lt_gt_contravar (sin x) (-1) H);
    change (-1) with (-(1));
    rewrite Ropp_involutive; clear H; intro;
      generalize
        (Rsqr_incrst_1 1 (- sin x) H (Rlt_le 0 1 Rlt_0_1)
          (Rlt_le 0 (- sin x) (Rlt_trans 0 1 (- sin x) Rlt_0_1 H)));
        rewrite Rsqr_1; intro; rewrite <- Rsqr_neg in H0;
          rewrite sin2 in H0; unfold Rminus in H0;
            generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0);
              rewrite <- Rplus_assoc; change (-1) with (-(1)); rewrite Rplus_opp_l;
                rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1;
                  generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1);
                    repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x));
                      intro; elim (Rlt_irrefl 0 (Rle_lt_trans 0 (Rsqr (cos x)) 0 H3 H2)).x:R
Hnle:~ -1 <= sin x
sin x < -1
  auto with real.
Qed.

Lemma COS_bound : forall x:R, -1 <= cos x <= 1.forall x : R, -1 <= cos x <= 1
Proof.forall x : R, -1 <= cos x <= 1
  intro; rewrite <- sin_shift; apply SIN_bound.
Qed.

Lemma cos_sin_0 : forall x:R, ~ (cos x = 0 /\ sin x = 0).forall x : R, ~ (cos x = 0 /\ sin x = 0)
Proof.forall x : R, ~ (cos x = 0 /\ sin x = 0)
  intro; red in |- *; intro; elim H; intros; generalize (sin2_cos2 x); intro;
    rewrite H0 in H2; rewrite H1 in H2; repeat rewrite Rsqr_0 in H2;
      rewrite Rplus_0_r in H2; generalize Rlt_0_1; intro;
        rewrite <- H2 in H3; elim (Rlt_irrefl 0 H3).
Qed.

Lemma cos_sin_0_var : forall x:R, cos x <> 0 \/ sin x <> 0.forall x : R, cos x <> 0 \/ sin x <> 0
Proof.forall x : R, cos x <> 0 \/ sin x <> 0
  intros x.x:R
cos x <> 0 \/ sin x <> 0
  destruct (Req_dec (cos x) 0).x:R
H:cos x = 0
cos x <> 0 \/ sin x <> 0
x:R
H:cos x <> 0
cos x <> 0 \/ sin x <> 0 2: now left.x:R
H:cos x = 0
cos x <> 0 \/ sin x <> 0
  right.x:R
H:cos x = 0
sin x <> 0 intros H'.x:R
H:cos x = 0
H':sin x = 0
False
  apply (cos_sin_0 x).x:R
H:cos x = 0
H':sin x = 0
cos x = 0 /\ sin x = 0
  now split.
Qed.

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

Using series definitions of cos and sin

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

Definition sin_lb (a:R) : R := sin_approx a 3.
Definition sin_ub (a:R) : R := sin_approx a 4.
Definition cos_lb (a:R) : R := cos_approx a 3.
Definition cos_ub (a:R) : R := cos_approx a 4.

Lemma sin_lb_gt_0 : forall a:R, 0 < a -> a <= PI / 2 -> 0 < sin_lb a.forall a : R, 0 < a -> a <= PI / 2 -> 0 < sin_lb a
Proof.forall a : R, 0 < a -> a <= PI / 2 -> 0 < sin_lb a
  intros.a:R
H:0 < a
H0:a <= PI / 2
0 < sin_lb a
  unfold sin_lb in |- *; unfold sin_approx in |- *; unfold sin_term in |- *.a:R
H:0 < a
H0:a <= PI / 2
0 <
sum_f_R0
  (fun i : nat =>
   (-1) ^ i *
   (a ^ (2 * i + 1) / INR (fact (2 * i + 1)))) 3
  set (Un := fun i:nat => a ^ (2 * i + 1) / INR (fact (2 * i + 1))).a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
0 <
sum_f_R0
  (fun i : nat =>
   (-1) ^ i *
   (a ^ (2 * i + 1) / INR (fact (2 * i + 1)))) 3
  replace
  (sum_f_R0
    (fun i:nat => (-1) ^ i * (a ^ (2 * i + 1) / INR (fact (2 * i + 1)))) 3)
    with (sum_f_R0 (fun i:nat => (-1) ^ i * Un i) 3);
      [ idtac | apply sum_eq; intros; unfold Un in |- *; reflexivity ].a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
0 < sum_f_R0 (fun i : nat => (-1) ^ i * Un i) 3
  cut (forall n:nat, Un (S n) < Un n).a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
(forall n : nat, Un (S n) < Un n) ->
0 < sum_f_R0 (fun i : nat => (-1) ^ i * Un i) 3
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
forall n : nat, Un (S n) < Un n
  intro; simpl in |- *.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
H1:forall n : nat, Un (S n) < Un n
0 <
1 * Un 0%nat + -1 * 1 * Un 1%nat +
-1 * (-1 * 1) * Un 2%nat +
-1 * (-1 * (-1 * 1)) * Un 3%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
forall n : nat, Un (S n) < Un n
  repeat rewrite Rmult_1_l; repeat rewrite Rmult_1_r;
    replace (-1 * Un 1%nat) with (- Un 1%nat); [ idtac | ring ];
    replace (-1 * -1 * Un 2%nat) with (Un 2%nat); [ idtac | ring ];
    replace (-1 * (-1 * -1) * Un 3%nat) with (- Un 3%nat);
    [ idtac | ring ];
    replace (Un 0%nat + - Un 1%nat + Un 2%nat + - Un 3%nat) with
    (Un 0%nat - Un 1%nat + (Un 2%nat - Un 3%nat)); [ idtac | ring ].a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
H1:forall n : nat, Un (S n) < Un n
0 < Un 0%nat - Un 1%nat + (Un 2%nat - Un 3%nat)
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
forall n : nat, Un (S n) < Un n
  apply Rplus_lt_0_compat.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
H1:forall n : nat, Un (S n) < Un n
0 < Un 0%nat - Un 1%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
H1:forall n : nat, Un (S n) < Un n
0 < Un 2%nat - Un 3%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
forall n : nat, Un (S n) < Un n
  unfold Rminus in |- *; apply Rplus_lt_reg_l with (Un 1%nat);
    rewrite Rplus_0_r; rewrite (Rplus_comm (Un 1%nat));
      rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
        apply H1.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
H1:forall n : nat, Un (S n) < Un n
0 < Un 2%nat - Un 3%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
forall n : nat, Un (S n) < Un n
  unfold Rminus in |- *; apply Rplus_lt_reg_l with (Un 3%nat);
    rewrite Rplus_0_r; rewrite (Rplus_comm (Un 3%nat));
      rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
        apply H1.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
forall n : nat, Un (S n) < Un n
  intro; unfold Un in |- *.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
a ^ (2 * S n + 1) / INR (fact (2 * S n + 1)) <
a ^ (2 * n + 1) / INR (fact (2 * n + 1))
  cut ((2 * S n + 1)%nat = (2 * n + 1 + 2)%nat).a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat ->
a ^ (2 * S n + 1) / INR (fact (2 * S n + 1)) <
a ^ (2 * n + 1) / INR (fact (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  intro; rewrite H1.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
a ^ (2 * n + 1 + 2) / INR (fact (2 * n + 1 + 2)) <
a ^ (2 * n + 1) / INR (fact (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  rewrite pow_add; unfold Rdiv in |- *; rewrite Rmult_assoc;
    apply Rmult_lt_compat_l.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
0 < a ^ (2 * n + 1)
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
a ^ 2 * / INR (fact (2 * n + 1 + 2)) <
/ INR (fact (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply pow_lt; assumption.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
a ^ 2 * / INR (fact (2 * n + 1 + 2)) <
/ INR (fact (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  rewrite <- H1; apply Rmult_lt_reg_l with (INR (fact (2 * n + 1))).a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
0 < INR (fact (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) *
(a ^ 2 * / INR (fact (2 * S n + 1))) <
INR (fact (2 * n + 1)) * / INR (fact (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply lt_INR_0; apply neq_O_lt.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
0%nat <> fact (2 * n + 1)
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) *
(a ^ 2 * / INR (fact (2 * S n + 1))) <
INR (fact (2 * n + 1)) * / INR (fact (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  assert (H2 := fact_neq_0 (2 * n + 1)).a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
H2:fact (2 * n + 1) <> 0%nat
0%nat <> fact (2 * n + 1)
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) *
(a ^ 2 * / INR (fact (2 * S n + 1))) <
INR (fact (2 * n + 1)) * / INR (fact (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  red in |- *; intro; elim H2; symmetry  in |- *; assumption.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) *
(a ^ 2 * / INR (fact (2 * S n + 1))) <
INR (fact (2 * n + 1)) * / INR (fact (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  rewrite <- Rinv_r_sym.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) *
(a ^ 2 * / INR (fact (2 * S n + 1))) < 1
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply Rmult_lt_reg_l with (INR (fact (2 * S n + 1))).a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
0 < INR (fact (2 * S n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) *
(INR (fact (2 * n + 1)) *
 (a ^ 2 * / INR (fact (2 * S n + 1)))) <
INR (fact (2 * S n + 1)) * 1
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply lt_INR_0; apply neq_O_lt.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
0%nat <> fact (2 * S n + 1)
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) *
(INR (fact (2 * n + 1)) *
 (a ^ 2 * / INR (fact (2 * S n + 1)))) <
INR (fact (2 * S n + 1)) * 1
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  assert (H2 := fact_neq_0 (2 * S n + 1)).a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
H2:fact (2 * S n + 1) <> 0%nat
0%nat <> fact (2 * S n + 1)
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) *
(INR (fact (2 * n + 1)) *
 (a ^ 2 * / INR (fact (2 * S n + 1)))) <
INR (fact (2 * S n + 1)) * 1
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  red in |- *; intro; elim H2; symmetry  in |- *; assumption.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) *
(INR (fact (2 * n + 1)) *
 (a ^ 2 * / INR (fact (2 * S n + 1)))) <
INR (fact (2 * S n + 1)) * 1
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  rewrite (Rmult_comm (INR (fact (2 * S n + 1)))); repeat rewrite Rmult_assoc;
    rewrite <- Rinv_l_sym.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) * (a ^ 2 * 1) <
INR (fact (2 * S n + 1)) * 1
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  do 2 rewrite Rmult_1_r; apply Rle_lt_trans with (INR (fact (2 * n + 1)) * 4).a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) * a ^ 2 <=
INR (fact (2 * n + 1)) * 4
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) * 4 < INR (fact (2 * S n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply Rmult_le_compat_l.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
0 <= INR (fact (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
a ^ 2 <= 4
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) * 4 < INR (fact (2 * S n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply pos_INR.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
a ^ 2 <= 4
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) * 4 < INR (fact (2 * S n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  simpl in |- *; rewrite Rmult_1_r; change 4 with (Rsqr 2);
    apply Rsqr_incr_1.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
a <= 2
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
0 <= a
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
0 <= 2
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) * 4 < INR (fact (2 * S n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply Rle_trans with (PI / 2);
    [ assumption
      | unfold Rdiv in |- *; apply Rmult_le_reg_l with 2;
        [ prove_sup0
          | rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m;
            [ apply PI_4 | discrR ] ] ].a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
0 <= a
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
0 <= 2
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) * 4 < INR (fact (2 * S n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  left; assumption.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
0 <= 2
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) * 4 < INR (fact (2 * S n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  left; prove_sup0.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) * 4 < INR (fact (2 * S n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  rewrite H1; replace (2 * n + 1 + 2)%nat with (S (S (2 * n + 1))).a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) * 4 <
INR (fact (S (S (2 * n + 1))))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  do 2 rewrite fact_simpl; do 2 rewrite mult_INR.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) * 4 <
INR (S (S (2 * n + 1))) *
(INR (S (2 * n + 1)) * INR (fact (2 * n + 1)))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  repeat rewrite <- Rmult_assoc.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) * 4 <
INR (S (S (2 * n + 1))) * INR (S (2 * n + 1)) *
INR (fact (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  rewrite <- (Rmult_comm (INR (fact (2 * n + 1)))).a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) * 4 <
INR (fact (2 * n + 1)) *
(INR (S (S (2 * n + 1))) * INR (S (2 * n + 1)))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply Rmult_lt_compat_l.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
0 < INR (fact (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
4 < INR (S (S (2 * n + 1))) * INR (S (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply lt_INR_0; apply neq_O_lt.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
0%nat <> fact (2 * n + 1)
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
4 < INR (S (S (2 * n + 1))) * INR (S (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  assert (H2 := fact_neq_0 (2 * n + 1)).a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
H2:fact (2 * n + 1) <> 0%nat
0%nat <> fact (2 * n + 1)
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
4 < INR (S (S (2 * n + 1))) * INR (S (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  red in |- *; intro; elim H2; symmetry  in |- *; assumption.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
4 < INR (S (S (2 * n + 1))) * INR (S (2 * n + 1))
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR; set (x := INR n);
    unfold INR in |- *.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
x:=INR n:R
4 < ((1 + 1) * x + 1 + 1 + 1) * ((1 + 1) * x + 1 + 1)
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  replace (((1 + 1) * x + 1 + 1 + 1) * ((1 + 1) * x + 1 + 1)) with (4 * x * x + 10 * x + 6);
  [ idtac | ring ].a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
x:=INR n:R
4 < 4 * x * x + 10 * x + 6
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply Rplus_lt_reg_l with (-(4)); rewrite Rplus_opp_l;
    replace (-(4) + (4 * x * x + 10 * x + 6)) with (4 * x * x + 10 * x + 2);
    [ idtac | ring ].a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
x:=INR n:R
0 < 4 * x * x + 10 * x + 2
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply Rplus_le_lt_0_compat.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
x:=INR n:R
0 <= 4 * x * x + 10 * x
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
x:=INR n:R
0 < 2
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  cut (0 <= x).a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
x:=INR n:R
0 <= x -> 0 <= 4 * x * x + 10 * x
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
x:=INR n:R
0 <= x
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
x:=INR n:R
0 < 2
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  intro; apply Rplus_le_le_0_compat; repeat apply Rmult_le_pos;
    assumption || left; prove_sup.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
x:=INR n:R
0 <= x
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
x:=INR n:R
0 < 2
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply pos_INR.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
x:=INR n:R
0 < 2
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  now apply IZR_lt.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
S (S (2 * n + 1)) = (2 * n + 1 + 2)%nat
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  ring.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * S n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply INR_fact_neq_0.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
H1:(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
INR (fact (2 * n + 1)) <> 0
a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  apply INR_fact_neq_0.a:R
H:0 < a
H0:a <= PI / 2
Un:=fun i : nat =>
a ^ (2 * i + 1) / INR (fact (2 * i + 1)):nat -> R
n:nat
(2 * S n + 1)%nat = (2 * n + 1 + 2)%nat
  ring.
Qed.

Lemma SIN : forall a:R, 0 <= a -> a <= PI -> sin_lb a <= sin a <= sin_ub a.forall a : R,
0 <= a -> a <= PI -> sin_lb a <= sin a <= sin_ub a
Proof.forall a : R,
0 <= a -> a <= PI -> sin_lb a <= sin a <= sin_ub a
  intros; unfold sin_lb, sin_ub in |- *; apply (sin_bound a 1 H H0).
Qed.

Lemma COS :
  forall a:R, - PI / 2 <= a -> a <= PI / 2 -> cos_lb a <= cos a <= cos_ub a.forall a : R,
- PI / 2 <= a ->
a <= PI / 2 -> cos_lb a <= cos a <= cos_ub a
Proof.forall a : R,
- PI / 2 <= a ->
a <= PI / 2 -> cos_lb a <= cos a <= cos_ub a
  intros; unfold cos_lb, cos_ub in |- *; apply (cos_bound a 1 H H0).
Qed.

(**********)
Lemma _PI2_RLT_0 : - (PI / 2) < 0.- (PI / 2) < 0
Proof.- (PI / 2) < 0
  assert (H := PI_RGT_0).H:PI > 0
- (PI / 2) < 0
  lra.
Qed.

Lemma PI4_RLT_PI2 : PI / 4 < PI / 2.PI / 4 < PI / 2
Proof.PI / 4 < PI / 2
  assert (H := PI_RGT_0).H:PI > 0
PI / 4 < PI / 2
  lra.
Qed.

Lemma PI2_Rlt_PI : PI / 2 < PI.PI / 2 < PI
Proof.PI / 2 < PI
  assert (H := PI_RGT_0).H:PI > 0
PI / 2 < PI
  lra.
Qed.

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

Increasing and decreasing of cos and sin

(***************************************************)
Theorem sin_gt_0 : forall x:R, 0 < x -> x < PI -> 0 < sin x.forall x : R, 0 < x -> x < PI -> 0 < sin x
Proof.forall x : R, 0 < x -> x < PI -> 0 < sin x
  intros; elim (SIN x (Rlt_le 0 x H) (Rlt_le x PI H0)); intros H1 _;
    case (Rtotal_order x (PI / 2)); intro H2.x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x < PI / 2
0 < sin x
x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
0 < sin x
  apply Rlt_le_trans with (sin_lb x).x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x < PI / 2
0 < sin_lb x
x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x < PI / 2
sin_lb x <= sin x
x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
0 < sin x
  apply sin_lb_gt_0; [ assumption | left; assumption ].x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x < PI / 2
sin_lb x <= sin x
x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
0 < sin x
  assumption.x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
0 < sin x
  elim H2; intro H3.x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
H3:x = PI / 2
0 < sin x
x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
H3:x > PI / 2
0 < sin x
  rewrite H3; rewrite sin_PI2; apply Rlt_0_1.x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
H3:x > PI / 2
0 < sin x
  rewrite <- sin_PI_x; generalize (Ropp_gt_lt_contravar x (PI / 2) H3);
    intro H4; generalize (Rplus_lt_compat_l PI (- x) (- (PI / 2)) H4).x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
H3:x > PI / 2
H4:- x < - (PI / 2)
PI + - x < PI + - (PI / 2) -> 0 < sin (PI - x)
  replace (PI + - (PI / 2)) with (PI / 2).x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
H3:x > PI / 2
H4:- x < - (PI / 2)
PI + - x < PI / 2 -> 0 < sin (PI - x)
x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
H3:x > PI / 2
H4:- x < - (PI / 2)
PI / 2 = PI + - (PI / 2)
  intro H5; generalize (Ropp_lt_gt_contravar x PI H0); intro H6;
    change (- PI < - x) in H6; generalize (Rplus_lt_compat_l PI (- PI) (- x) H6).x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
H3:x > PI / 2
H4:- x < - (PI / 2)
H5:PI + - x < PI / 2
H6:- PI < - x
PI + - PI < PI + - x -> 0 < sin (PI - x)
x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
H3:x > PI / 2
H4:- x < - (PI / 2)
PI / 2 = PI + - (PI / 2)
  rewrite Rplus_opp_r.x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
H3:x > PI / 2
H4:- x < - (PI / 2)
H5:PI + - x < PI / 2
H6:- PI < - x
0 < PI + - x -> 0 < sin (PI - x)
x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
H3:x > PI / 2
H4:- x < - (PI / 2)
PI / 2 = PI + - (PI / 2)
  intro H7;
    elim
      (SIN (PI - x) (Rlt_le 0 (PI - x) H7)
        (Rlt_le (PI - x) PI (Rlt_trans (PI - x) (PI / 2) PI H5 PI2_Rlt_PI)));
      intros H8 _;
        generalize (sin_lb_gt_0 (PI - x) H7 (Rlt_le (PI - x) (PI / 2) H5));
          intro H9; apply (Rlt_le_trans 0 (sin_lb (PI - x)) (sin (PI - x)) H9 H8).x:R
H:0 < x
H0:x < PI
H1:sin_lb x <= sin x
H2:x = PI / 2 \/ x > PI / 2
H3:x > PI / 2
H4:- x < - (PI / 2)
PI / 2 = PI + - (PI / 2)
  field.
Qed.

Theorem cos_gt_0 : forall x:R, - (PI / 2) < x -> x < PI / 2 -> 0 < cos x.forall x : R,
- (PI / 2) < x -> x < PI / 2 -> 0 < cos x
Proof.forall x : R,
- (PI / 2) < x -> x < PI / 2 -> 0 < cos x
  intros; rewrite cos_sin;
    generalize (Rplus_lt_compat_l (PI / 2) (- (PI / 2)) x H).x:R
H:- (PI / 2) < x
H0:x < PI / 2
PI / 2 + - (PI / 2) < PI / 2 + x ->
0 < sin (PI / 2 + x)
  rewrite Rplus_opp_r; intro H1;
    generalize (Rplus_lt_compat_l (PI / 2) x (PI / 2) H0);
      rewrite <- double_var; intro H2; apply (sin_gt_0 (PI / 2 + x) H1 H2).
Qed.

Lemma sin_ge_0 : forall x:R, 0 <= x -> x <= PI -> 0 <= sin x.forall x : R, 0 <= x -> x <= PI -> 0 <= sin x
Proof.forall x : R, 0 <= x -> x <= PI -> 0 <= sin x
  intros x H1 H2; elim H1; intro H3;
    [ elim H2; intro H4;
      [ left; apply (sin_gt_0 x H3 H4)
        | rewrite H4; right; symmetry  in |- *; apply sin_PI ]
      | rewrite <- H3; right; symmetry  in |- *; apply sin_0 ].
Qed.

Lemma cos_ge_0 : forall x:R, - (PI / 2) <= x -> x <= PI / 2 -> 0 <= cos x.forall x : R,
- (PI / 2) <= x -> x <= PI / 2 -> 0 <= cos x
Proof.forall x : R,
- (PI / 2) <= x -> x <= PI / 2 -> 0 <= cos x
  intros x H1 H2; elim H1; intro H3;
    [ elim H2; intro H4;
      [ left; apply (cos_gt_0 x H3 H4)
        | rewrite H4; right; symmetry  in |- *; apply cos_PI2 ]
      | rewrite <- H3; rewrite cos_neg; right; symmetry  in |- *; apply cos_PI2 ].
Qed.

Lemma sin_le_0 : forall x:R, PI <= x -> x <= 2 * PI -> sin x <= 0.forall x : R, PI <= x -> x <= 2 * PI -> sin x <= 0
Proof.forall x : R, PI <= x -> x <= 2 * PI -> sin x <= 0
  intros x H1 H2; apply Rge_le; rewrite <- Ropp_0;
    rewrite <- (Ropp_involutive (sin x)); apply Ropp_le_ge_contravar;
      rewrite <- neg_sin; replace (x + PI) with (x - PI + 2 * INR 1 * PI);
        [ rewrite (sin_period (x - PI) 1); apply sin_ge_0;
          [ replace (x - PI) with (x + - PI);
            [ rewrite Rplus_comm; replace 0 with (- PI + PI);
              [ apply Rplus_le_compat_l; assumption | ring ]
              | ring ]
            | replace (x - PI) with (x + - PI); rewrite Rplus_comm;
              [ pattern PI at 2 in |- *; replace PI with (- PI + 2 * PI);
                [ apply Rplus_le_compat_l; assumption | ring ]
                | ring ] ]
          | unfold INR in |- *; ring ].
Qed.

Lemma cos_le_0 : forall x:R, PI / 2 <= x -> x <= 3 * (PI / 2) -> cos x <= 0.forall x : R,
PI / 2 <= x -> x <= 3 * (PI / 2) -> cos x <= 0
Proof.forall x : R,
PI / 2 <= x -> x <= 3 * (PI / 2) -> cos x <= 0
  intros x H1 H2; apply Rge_le; rewrite <- Ropp_0;
    rewrite <- (Ropp_involutive (cos x)); apply Ropp_le_ge_contravar;
      rewrite <- neg_cos; replace (x + PI) with (x - PI + 2 * INR 1 * PI).x:R
H1:PI / 2 <= x
H2:x <= 3 * (PI / 2)
0 <= cos (x - PI + 2 * INR 1 * PI)
x:R
H1:PI / 2 <= x
H2:x <= 3 * (PI / 2)
x - PI + 2 * INR 1 * PI = x + PI
  rewrite cos_period; apply cos_ge_0.x:R
H1:PI / 2 <= x
H2:x <= 3 * (PI / 2)
- (PI / 2) <= x - PI
x:R
H1:PI / 2 <= x
H2:x <= 3 * (PI / 2)
x - PI <= PI / 2
x:R
H1:PI / 2 <= x
H2:x <= 3 * (PI / 2)
x - PI + 2 * INR 1 * PI = x + PI
  replace (- (PI / 2)) with (- PI + PI / 2) by field.x:R
H1:PI / 2 <= x
H2:x <= 3 * (PI / 2)
- PI + PI / 2 <= x - PI
x:R
H1:PI / 2 <= x
H2:x <= 3 * (PI / 2)
x - PI <= PI / 2
x:R
H1:PI / 2 <= x
H2:x <= 3 * (PI / 2)
x - PI + 2 * INR 1 * PI = x + PI
  unfold Rminus in |- *; rewrite (Rplus_comm x); apply Rplus_le_compat_l;
    assumption.x:R
H1:PI / 2 <= x
H2:x <= 3 * (PI / 2)
x - PI <= PI / 2
x:R
H1:PI / 2 <= x
H2:x <= 3 * (PI / 2)
x - PI + 2 * INR 1 * PI = x + PI
  unfold Rminus in |- *; rewrite Rplus_comm;
    replace (PI / 2) with (- PI + 3 * (PI / 2)) by field.x:R
H1:PI / 2 <= x
H2:x <= 3 * (PI / 2)
- PI + x <= - PI + 3 * (PI / 2)
x:R
H1:PI / 2 <= x
H2:x <= 3 * (PI / 2)
x - PI + 2 * INR 1 * PI = x + PI
  apply Rplus_le_compat_l; assumption.x:R
H1:PI / 2 <= x
H2:x <= 3 * (PI / 2)
x - PI + 2 * INR 1 * PI = x + PI
  unfold INR in |- *; ring.
Qed.

Lemma sin_lt_0 : forall x:R, PI < x -> x < 2 * PI -> sin x < 0.forall x : R, PI < x -> x < 2 * PI -> sin x < 0
Proof.forall x : R, PI < x -> x < 2 * PI -> sin x < 0
  intros x H1 H2; rewrite <- Ropp_0; rewrite <- (Ropp_involutive (sin x));
    apply Ropp_lt_gt_contravar; rewrite <- neg_sin;
      replace (x + PI) with (x - PI + 2 * INR 1 * PI);
      [ rewrite (sin_period (x - PI) 1); apply sin_gt_0;
        [ replace (x - PI) with (x + - PI);
          [ rewrite Rplus_comm; replace 0 with (- PI + PI);
            [ apply Rplus_lt_compat_l; assumption | ring ]
            | ring ]
          | replace (x - PI) with (x + - PI); rewrite Rplus_comm;
            [ pattern PI at 2 in |- *; replace PI with (- PI + 2 * PI);
              [ apply Rplus_lt_compat_l; assumption | ring ]
              | ring ] ]
        | unfold INR in |- *; ring ].
Qed.

Lemma sin_lt_0_var : forall x:R, - PI < x -> x < 0 -> sin x < 0.forall x : R, - PI < x -> x < 0 -> sin x < 0
Proof.forall x : R, - PI < x -> x < 0 -> sin x < 0
  intros; generalize (Rplus_lt_compat_l (2 * PI) (- PI) x H);
    replace (2 * PI + - PI) with PI;
    [ intro H1; rewrite Rplus_comm in H1;
      generalize (Rplus_lt_compat_l (2 * PI) x 0 H0);
        intro H2; rewrite (Rplus_comm (2 * PI)) in H2;
          rewrite <- (Rplus_comm 0) in H2; rewrite Rplus_0_l in H2;
            rewrite <- (sin_period x 1); unfold INR in |- *;
              replace (2 * 1 * PI) with (2 * PI);
              [ apply (sin_lt_0 (x + 2 * PI) H1 H2) | ring ]
      | ring ].
Qed.

Lemma cos_lt_0 : forall x:R, PI / 2 < x -> x < 3 * (PI / 2) -> cos x < 0.forall x : R,
PI / 2 < x -> x < 3 * (PI / 2) -> cos x < 0
Proof.forall x : R,
PI / 2 < x -> x < 3 * (PI / 2) -> cos x < 0
  intros x H1 H2; rewrite <- Ropp_0; rewrite <- (Ropp_involutive (cos x));
    apply Ropp_lt_gt_contravar; rewrite <- neg_cos;
      replace (x + PI) with (x - PI + 2 * INR 1 * PI).x:R
H1:PI / 2 < x
H2:x < 3 * (PI / 2)
0 < cos (x - PI + 2 * INR 1 * PI)
x:R
H1:PI / 2 < x
H2:x < 3 * (PI / 2)
x - PI + 2 * INR 1 * PI = x + PI
  rewrite cos_period; apply cos_gt_0.x:R
H1:PI / 2 < x
H2:x < 3 * (PI / 2)
- (PI / 2) < x - PI
x:R
H1:PI / 2 < x
H2:x < 3 * (PI / 2)
x - PI < PI / 2
x:R
H1:PI / 2 < x
H2:x < 3 * (PI / 2)
x - PI + 2 * INR 1 * PI = x + PI
  replace (- (PI / 2)) with (- PI + PI / 2) by field.x:R
H1:PI / 2 < x
H2:x < 3 * (PI / 2)
- PI + PI / 2 < x - PI
x:R
H1:PI / 2 < x
H2:x < 3 * (PI / 2)
x - PI < PI / 2
x:R
H1:PI / 2 < x
H2:x < 3 * (PI / 2)
x - PI + 2 * INR 1 * PI = x + PI
  unfold Rminus in |- *; rewrite (Rplus_comm x); apply Rplus_lt_compat_l;
    assumption.x:R
H1:PI / 2 < x
H2:x < 3 * (PI / 2)
x - PI < PI / 2
x:R
H1:PI / 2 < x
H2:x < 3 * (PI / 2)
x - PI + 2 * INR 1 * PI = x + PI
  unfold Rminus in |- *; rewrite Rplus_comm;
    replace (PI / 2) with (- PI + 3 * (PI / 2)) by field.x:R
H1:PI / 2 < x
H2:x < 3 * (PI / 2)
- PI + x < - PI + 3 * (PI / 2)
x:R
H1:PI / 2 < x
H2:x < 3 * (PI / 2)
x - PI + 2 * INR 1 * PI = x + PI
  apply Rplus_lt_compat_l; assumption.x:R
H1:PI / 2 < x
H2:x < 3 * (PI / 2)
x - PI + 2 * INR 1 * PI = x + PI
  unfold INR in |- *; ring.
Qed.

Lemma tan_gt_0 : forall x:R, 0 < x -> x < PI / 2 -> 0 < tan x.forall x : R, 0 < x -> x < PI / 2 -> 0 < tan x
Proof.forall x : R, 0 < x -> x < PI / 2 -> 0 < tan x
  intros x H1 H2; unfold tan in |- *; generalize _PI2_RLT_0;
    generalize (Rlt_trans 0 x (PI / 2) H1 H2); intros;
      generalize (Rlt_trans (- (PI / 2)) 0 x H0 H1); intro H5;
        generalize (Rlt_trans x (PI / 2) PI H2 PI2_Rlt_PI);
          intro H7; unfold Rdiv in |- *; apply Rmult_lt_0_compat.x:R
H1:0 < x
H2:x < PI / 2
H:0 < PI / 2
H0:- (PI / 2) < 0
H5:- (PI / 2) < x
H7:x < PI
0 < sin x
x:R
H1:0 < x
H2:x < PI / 2
H:0 < PI / 2
H0:- (PI / 2) < 0
H5:- (PI / 2) < x
H7:x < PI
0 < / cos x
  apply sin_gt_0; assumption.x:R
H1:0 < x
H2:x < PI / 2
H:0 < PI / 2
H0:- (PI / 2) < 0
H5:- (PI / 2) < x
H7:x < PI
0 < / cos x
  apply Rinv_0_lt_compat; apply cos_gt_0; assumption.
Qed.

Lemma tan_lt_0 : forall x:R, - (PI / 2) < x -> x < 0 -> tan x < 0.forall x : R, - (PI / 2) < x -> x < 0 -> tan x < 0
Proof.forall x : R, - (PI / 2) < x -> x < 0 -> tan x < 0
  intros x H1 H2; unfold tan in |- *;
    generalize (cos_gt_0 x H1 (Rlt_trans x 0 (PI / 2) H2 PI2_RGT_0));
      intro H3; rewrite <- Ropp_0;
        replace (sin x / cos x) with (- (- sin x / cos x)).x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
- (- sin x / cos x) < - 0
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
- (- sin x / cos x) = sin x / cos x
  rewrite <- sin_neg; apply Ropp_gt_lt_contravar;
    change (0 < sin (- x) / cos x) in |- *; unfold Rdiv in |- *;
      apply Rmult_lt_0_compat.x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
0 < sin (- x)
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
0 < / cos x
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
- (- sin x / cos x) = sin x / cos x
  apply sin_gt_0.x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
0 < - x
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
- x < PI
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
0 < / cos x
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
- (- sin x / cos x) = sin x / cos x
  rewrite <- Ropp_0; apply Ropp_gt_lt_contravar; assumption.x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
- x < PI
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
0 < / cos x
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
- (- sin x / cos x) = sin x / cos x
  apply Rlt_trans with (PI / 2).x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
- x < PI / 2
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
PI / 2 < PI
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
0 < / cos x
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
- (- sin x / cos x) = sin x / cos x
  rewrite <- (Ropp_involutive (PI / 2)); apply Ropp_gt_lt_contravar; assumption.x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
PI / 2 < PI
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
0 < / cos x
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
- (- sin x / cos x) = sin x / cos x
  apply PI2_Rlt_PI.x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
0 < / cos x
x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
- (- sin x / cos x) = sin x / cos x
  apply Rinv_0_lt_compat; assumption.x:R
H1:- (PI / 2) < x
H2:x < 0
H3:0 < cos x
- (- sin x / cos x) = sin x / cos x
  unfold Rdiv in |- *; ring.
Qed.

Lemma cos_ge_0_3PI2 :
  forall x:R, 3 * (PI / 2) <= x -> x <= 2 * PI -> 0 <= cos x.forall x : R,
3 * (PI / 2) <= x -> x <= 2 * PI -> 0 <= cos x
Proof.forall x : R,
3 * (PI / 2) <= x -> x <= 2 * PI -> 0 <= cos x
  intros; rewrite <- cos_neg; rewrite <- (cos_period (- x) 1);
    unfold INR in |- *; replace (- x + 2 * 1 * PI) with (2 * PI - x) by ring.x:R
H:3 * (PI / 2) <= x
H0:x <= 2 * PI
0 <= cos (2 * PI - x)
  generalize (Ropp_le_ge_contravar x (2 * PI) H0); intro H1;
    generalize (Rge_le (- x) (- (2 * PI)) H1); clear H1;
      intro H1; generalize (Rplus_le_compat_l (2 * PI) (- (2 * PI)) (- x) H1).x:R
H:3 * (PI / 2) <= x
H0:x <= 2 * PI
H1:- (2 * PI) <= - x
2 * PI + - (2 * PI) <= 2 * PI + - x ->
0 <= cos (2 * PI - x)
  rewrite Rplus_opp_r.x:R
H:3 * (PI / 2) <= x
H0:x <= 2 * PI
H1:- (2 * PI) <= - x
0 <= 2 * PI + - x -> 0 <= cos (2 * PI - x)
  intro H2; generalize (Ropp_le_ge_contravar (3 * (PI / 2)) x H); intro H3;
    generalize (Rge_le (- (3 * (PI / 2))) (- x) H3); clear H3;
      intro H3;
        generalize (Rplus_le_compat_l (2 * PI) (- x) (- (3 * (PI / 2))) H3).x:R
H:3 * (PI / 2) <= x
H0:x <= 2 * PI
H1:- (2 * PI) <= - x
H2:0 <= 2 * PI + - x
H3:- x <= - (3 * (PI / 2))
2 * PI + - x <= 2 * PI + - (3 * (PI / 2)) ->
0 <= cos (2 * PI - x)
  replace (2 * PI + - (3 * (PI / 2))) with (PI / 2) by field.x:R
H:3 * (PI / 2) <= x
H0:x <= 2 * PI
H1:- (2 * PI) <= - x
H2:0 <= 2 * PI + - x
H3:- x <= - (3 * (PI / 2))
2 * PI + - x <= PI / 2 -> 0 <= cos (2 * PI - x)
  intro H4;
    apply
      (cos_ge_0 (2 * PI - x)
        (Rlt_le (- (PI / 2)) (2 * PI - x)
          (Rlt_le_trans (- (PI / 2)) 0 (2 * PI - x) _PI2_RLT_0 H2)) H4).
Qed.

Lemma form1 :
  forall p q:R, cos p + cos q = 2 * cos ((p - q) / 2) * cos ((p + q) / 2).forall p q : R,
cos p + cos q =
2 * cos ((p - q) / 2) * cos ((p + q) / 2)
Proof.forall p q : R,
cos p + cos q =
2 * cos ((p - q) / 2) * cos ((p + q) / 2)
  intros p q; pattern p at 1 in |- *;
    replace p with ((p - q) / 2 + (p + q) / 2) by field.p, q:R
cos ((p - q) / 2 + (p + q) / 2) + cos q =
2 * cos ((p - q) / 2) * cos ((p + q) / 2)
  rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2) by field.p, q:R
cos ((p - q) / 2 + (p + q) / 2) +
cos ((p - q) / 2 - (p + q) / 2) =
2 * cos ((p - q) / 2) * cos ((p + q) / 2)
  rewrite cos_plus; rewrite cos_minus; ring.
Qed.

Lemma form2 :
  forall p q:R, cos p - cos q = -2 * sin ((p - q) / 2) * sin ((p + q) / 2).forall p q : R,
cos p - cos q =
-2 * sin ((p - q) / 2) * sin ((p + q) / 2)
Proof.forall p q : R,
cos p - cos q =
-2 * sin ((p - q) / 2) * sin ((p + q) / 2)
  intros p q; pattern p at 1 in |- *;
    replace p with ((p - q) / 2 + (p + q) / 2) by field.p, q:R
cos ((p - q) / 2 + (p + q) / 2) - cos q =
-2 * sin ((p - q) / 2) * sin ((p + q) / 2)
  rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2) by field.p, q:R
cos ((p - q) / 2 + (p + q) / 2) -
cos ((p - q) / 2 - (p + q) / 2) =
-2 * sin ((p - q) / 2) * sin ((p + q) / 2)
  rewrite cos_plus; rewrite cos_minus; ring.
Qed.

Lemma form3 :
  forall p q:R, sin p + sin q = 2 * cos ((p - q) / 2) * sin ((p + q) / 2).forall p q : R,
sin p + sin q =
2 * cos ((p - q) / 2) * sin ((p + q) / 2)
Proof.forall p q : R,
sin p + sin q =
2 * cos ((p - q) / 2) * sin ((p + q) / 2)
  intros p q; pattern p at 1 in |- *;
    replace p with ((p - q) / 2 + (p + q) / 2).p, q:R
sin ((p - q) / 2 + (p + q) / 2) + sin q =
2 * cos ((p - q) / 2) * sin ((p + q) / 2)
p, q:R
(p - q) / 2 + (p + q) / 2 = p
  pattern q at 3 in |- *; replace q with ((p + q) / 2 - (p - q) / 2).p, q:R
sin ((p - q) / 2 + (p + q) / 2) +
sin ((p + q) / 2 - (p - q) / 2) =
2 * cos ((p - q) / 2) * sin ((p + q) / 2)
p, q:R
(p + q) / 2 - (p - q) / 2 = q
p, q:R
(p - q) / 2 + (p + q) / 2 = p
  rewrite sin_plus; rewrite sin_minus; ring.p, q:R
(p + q) / 2 - (p - q) / 2 = q
p, q:R
(p - q) / 2 + (p + q) / 2 = p
  pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.p, q:R
(p - q) / 2 + (p + q) / 2 = p
  pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
Qed.

Lemma form4 :
  forall p q:R, sin p - sin q = 2 * cos ((p + q) / 2) * sin ((p - q) / 2).forall p q : R,
sin p - sin q =
2 * cos ((p + q) / 2) * sin ((p - q) / 2)
Proof.forall p q : R,
sin p - sin q =
2 * cos ((p + q) / 2) * sin ((p - q) / 2)
  intros p q; pattern p at 1 in |- *;
    replace p with ((p - q) / 2 + (p + q) / 2) by field.p, q:R
sin ((p - q) / 2 + (p + q) / 2) - sin q =
2 * cos ((p + q) / 2) * sin ((p - q) / 2)
  pattern q at 3 in |- *; replace q with ((p + q) / 2 - (p - q) / 2) by field.p, q:R
sin ((p - q) / 2 + (p + q) / 2) -
sin ((p + q) / 2 - (p - q) / 2) =
2 * cos ((p + q) / 2) * sin ((p - q) / 2)
  rewrite sin_plus; rewrite sin_minus; ring.

Qed.

Lemma sin_increasing_0 :
  forall x y:R,
    - (PI / 2) <= x ->
    x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> sin x < sin y -> x < y.forall x y : R,
- (PI / 2) <= x ->
x <= PI / 2 ->
- (PI / 2) <= y ->
y <= PI / 2 -> sin x < sin y -> x < y
Proof.forall x y : R,
- (PI / 2) <= x ->
x <= PI / 2 ->
- (PI / 2) <= y ->
y <= PI / 2 -> sin x < sin y -> x < y
  intros; cut (sin ((x - y) / 2) < 0).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0 -> x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  intro H4; case (Rtotal_order ((x - y) / 2) 0); intro H5.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 < 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  assert (Hyp : 0 < 2).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 < 0
0 < 2
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 < 0
Hyp:0 < 2
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  prove_sup0.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 < 0
Hyp:0 < 2
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  generalize (Rmult_lt_compat_l 2 ((x - y) / 2) 0 Hyp H5).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 < 0
Hyp:0 < 2
2 * ((x - y) / 2) < 2 * 0 -> x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  unfold Rdiv in |- *.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 < 0
Hyp:0 < 2
2 * ((x - y) * / 2) < 2 * 0 -> x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  rewrite <- Rmult_assoc.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 < 0
Hyp:0 < 2
2 * (x - y) * / 2 < 2 * 0 -> x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  rewrite Rinv_r_simpl_m.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 < 0
Hyp:0 < 2
x - y < 2 * 0 -> x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 < 0
Hyp:0 < 2
2 <> 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  rewrite Rmult_0_r.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 < 0
Hyp:0 < 2
x - y < 0 -> x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 < 0
Hyp:0 < 2
2 <> 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  clear H5; intro H5; apply Rminus_lt; assumption.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 < 0
Hyp:0 < 2
2 <> 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  discrR.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  elim H5; intro H6.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
H6:(x - y) / 2 = 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
H6:(x - y) / 2 > 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  rewrite H6 in H4; rewrite sin_0 in H4; elim (Rlt_irrefl 0 H4).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
H6:(x - y) / 2 > 0
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  change (0 < (x - y) / 2) in H6;
    generalize (Ropp_le_ge_contravar (- (PI / 2)) y H1).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
H6:0 < (x - y) / 2
- - (PI / 2) >= - y -> x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  rewrite Ropp_involutive.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
H6:0 < (x - y) / 2
PI / 2 >= - y -> x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  intro H7; generalize (Rge_le (PI / 2) (- y) H7); clear H7; intro H7;
    generalize (Rplus_le_compat x (PI / 2) (- y) (PI / 2) H0 H7).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
H6:0 < (x - y) / 2
H7:- y <= PI / 2
x + - y <= PI / 2 + PI / 2 -> x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  rewrite <- double_var.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
H6:0 < (x - y) / 2
H7:- y <= PI / 2
x + - y <= PI -> x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  intro H8.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
H6:0 < (x - y) / 2
H7:- y <= PI / 2
H8:x + - y <= PI
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  assert (Hyp : 0 < 2).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
H6:0 < (x - y) / 2
H7:- y <= PI / 2
H8:x + - y <= PI
0 < 2
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
H6:0 < (x - y) / 2
H7:- y <= PI / 2
H8:x + - y <= PI
Hyp:0 < 2
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  prove_sup0.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
H6:0 < (x - y) / 2
H7:- y <= PI / 2
H8:x + - y <= PI
Hyp:0 < 2
x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  generalize
    (Rmult_le_compat_l (/ 2) (x - y) PI
      (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H8).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
H6:0 < (x - y) / 2
H7:- y <= PI / 2
H8:x + - y <= PI
Hyp:0 < 2
/ 2 * (x - y) <= / 2 * PI -> x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  repeat rewrite (Rmult_comm (/ 2)).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
H4:sin ((x - y) / 2) < 0
H5:(x - y) / 2 = 0 \/ (x - y) / 2 > 0
H6:0 < (x - y) / 2
H7:- y <= PI / 2
H8:x + - y <= PI
Hyp:0 < 2
(x - y) * / 2 <= PI * / 2 -> x < y
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  intro H9;
    generalize
      (sin_gt_0 ((x - y) / 2) H6
        (Rle_lt_trans ((x - y) / 2) (PI / 2) PI H9 PI2_Rlt_PI));
      intro H10;
        elim
          (Rlt_irrefl (sin ((x - y) / 2))
            (Rlt_trans (sin ((x - y) / 2)) 0 (sin ((x - y) / 2)) H4 H10)).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:sin x < sin y
sin ((x - y) / 2) < 0
  generalize (Rlt_minus (sin x) (sin y) H3); clear H3; intro H3;
    rewrite form4 in H3;
      generalize (Rplus_le_compat x (PI / 2) y (PI / 2) H0 H2).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
x + y <= PI / 2 + PI / 2 -> sin ((x - y) / 2) < 0
  rewrite <- double_var.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
x + y <= PI -> sin ((x - y) / 2) < 0
  assert (Hyp : 0 < 2).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
0 < 2
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
x + y <= PI -> sin ((x - y) / 2) < 0
  prove_sup0.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
x + y <= PI -> sin ((x - y) / 2) < 0
  intro H4;
    generalize
      (Rmult_le_compat_l (/ 2) (x + y) PI
        (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H4).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:x + y <= PI
/ 2 * (x + y) <= / 2 * PI -> sin ((x - y) / 2) < 0
  repeat rewrite (Rmult_comm (/ 2)).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:x + y <= PI
(x + y) * / 2 <= PI * / 2 -> sin ((x - y) / 2) < 0
  clear H4; intro H4;
    generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) y H H1);
      replace (- (PI / 2) + - (PI / 2)) with (- PI) by field.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
- PI <= x + y -> sin ((x - y) / 2) < 0
  intro H5;
    generalize
      (Rmult_le_compat_l (/ 2) (- PI) (x + y)
        (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H5).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- PI <= x + y
/ 2 * - PI <= / 2 * (x + y) -> sin ((x - y) / 2) < 0
  replace (/ 2 * (x + y)) with ((x + y) / 2) by apply Rmult_comm.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- PI <= x + y
/ 2 * - PI <= (x + y) / 2 -> sin ((x - y) / 2) < 0
  replace (/ 2 * - PI) with (- (PI / 2)) by field.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- PI <= x + y
- (PI / 2) <= (x + y) / 2 -> sin ((x - y) / 2) < 0
  clear H5; intro H5; elim H4; intro H40.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 < PI * / 2
sin ((x - y) / 2) < 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 = PI * / 2
sin ((x - y) / 2) < 0
  elim H5; intro H50.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 < PI * / 2
H50:- (PI / 2) < (x + y) / 2
sin ((x - y) / 2) < 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 < PI * / 2
H50:- (PI / 2) = (x + y) / 2
sin ((x - y) / 2) < 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 = PI * / 2
sin ((x - y) / 2) < 0
  generalize (cos_gt_0 ((x + y) / 2) H50 H40); intro H6;
    generalize (Rmult_lt_compat_l 2 0 (cos ((x + y) / 2)) Hyp H6).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 < PI * / 2
H50:- (PI / 2) < (x + y) / 2
H6:0 < cos ((x + y) / 2)
2 * 0 < 2 * cos ((x + y) / 2) -> sin ((x - y) / 2) < 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 < PI * / 2
H50:- (PI / 2) = (x + y) / 2
sin ((x - y) / 2) < 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 = PI * / 2
sin ((x - y) / 2) < 0
  rewrite Rmult_0_r.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 < PI * / 2
H50:- (PI / 2) < (x + y) / 2
H6:0 < cos ((x + y) / 2)
0 < 2 * cos ((x + y) / 2) -> sin ((x - y) / 2) < 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 < PI * / 2
H50:- (PI / 2) = (x + y) / 2
sin ((x - y) / 2) < 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 = PI * / 2
sin ((x - y) / 2) < 0
  clear H6; intro H6; case (Rcase_abs (sin ((x - y) / 2))); intro H7.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 < PI * / 2
H50:- (PI / 2) < (x + y) / 2
H6:0 < 2 * cos ((x + y) / 2)
H7:sin ((x - y) / 2) < 0
sin ((x - y) / 2) < 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 < PI * / 2
H50:- (PI / 2) < (x + y) / 2
H6:0 < 2 * cos ((x + y) / 2)
H7:sin ((x - y) / 2) >= 0
sin ((x - y) / 2) < 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 < PI * / 2
H50:- (PI / 2) = (x + y) / 2
sin ((x - y) / 2) < 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 = PI * / 2
sin ((x - y) / 2) < 0
  assumption.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 < PI * / 2
H50:- (PI / 2) < (x + y) / 2
H6:0 < 2 * cos ((x + y) / 2)
H7:sin ((x - y) / 2) >= 0
sin ((x - y) / 2) < 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 < PI * / 2
H50:- (PI / 2) = (x + y) / 2
sin ((x - y) / 2) < 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 = PI * / 2
sin ((x - y) / 2) < 0
  generalize (Rge_le (sin ((x - y) / 2)) 0 H7); clear H7; intro H7;
    generalize
      (Rmult_le_pos (2 * cos ((x + y) / 2)) (sin ((x - y) / 2))
        (Rlt_le 0 (2 * cos ((x + y) / 2)) H6) H7); intro H8;
      generalize
        (Rle_lt_trans 0 (2 * cos ((x + y) / 2) * sin ((x - y) / 2)) 0 H8 H3);
        intro H9; elim (Rlt_irrefl 0 H9).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 < PI * / 2
H50:- (PI / 2) = (x + y) / 2
sin ((x - y) / 2) < 0
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 = PI * / 2
sin ((x - y) / 2) < 0
  rewrite <- H50 in H3; rewrite cos_neg in H3; rewrite cos_PI2 in H3;
    rewrite Rmult_0_r in H3; rewrite Rmult_0_l in H3;
      elim (Rlt_irrefl 0 H3).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 = PI * / 2
sin ((x - y) / 2) < 0
  unfold Rdiv in H3.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:2 * cos ((x + y) * / 2) * sin ((x - y) * / 2) < 0
Hyp:0 < 2
H4:(x + y) * / 2 <= PI * / 2
H5:- (PI / 2) <= (x + y) / 2
H40:(x + y) * / 2 = PI * / 2
sin ((x - y) / 2) < 0
  rewrite H40 in H3; assert (H50 := cos_PI2); unfold Rdiv in H50;
    rewrite H50 in H3; rewrite Rmult_0_r in H3; rewrite Rmult_0_l in H3;
      elim (Rlt_irrefl 0 H3).
Qed.

Lemma sin_increasing_1 :
  forall x y:R,
    - (PI / 2) <= x ->
    x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> x < y -> sin x < sin y.forall x y : R,
- (PI / 2) <= x ->
x <= PI / 2 ->
- (PI / 2) <= y ->
y <= PI / 2 -> x < y -> sin x < sin y
Proof.forall x y : R,
- (PI / 2) <= x ->
x <= PI / 2 ->
- (PI / 2) <= y ->
y <= PI / 2 -> x < y -> sin x < sin y
  intros; generalize (Rplus_lt_compat_l x x y H3); intro H4;
    generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) x H H);
      replace (- (PI / 2) + - (PI / 2)) with (- PI) by field.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:x < y
H4:x + x < x + y
- PI <= x + x -> sin x < sin y
  assert (Hyp : 0 < 2).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:x < y
H4:x + x < x + y
0 < 2
x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:x < y
H4:x + x < x + y
Hyp:0 < 2
- PI <= x + x -> sin x < sin y
  prove_sup0.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:x < y
H4:x + x < x + y
Hyp:0 < 2
- PI <= x + x -> sin x < sin y
  intro H5; generalize (Rle_lt_trans (- PI) (x + x) (x + y) H5 H4); intro H6;
    generalize
      (Rmult_lt_compat_l (/ 2) (- PI) (x + y) (Rinv_0_lt_compat 2 Hyp) H6);
      replace (/ 2 * - PI) with (- (PI / 2)) by field.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:x < y
H4:x + x < x + y
Hyp:0 < 2
H5:- PI <= x + x
H6:- PI < x + y
- (PI / 2) < / 2 * (x + y) -> sin x < sin y
  replace (/ 2 * (x + y)) with ((x + y) / 2) by apply Rmult_comm.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:x < y
H4:x + x < x + y
Hyp:0 < 2
H5:- PI <= x + x
H6:- PI < x + y
- (PI / 2) < (x + y) / 2 -> sin x < sin y
  clear H4 H5 H6; intro H4; generalize (Rplus_lt_compat_l y x y H3); intro H5;
    rewrite Rplus_comm in H5;
      generalize (Rplus_le_compat y (PI / 2) y (PI / 2) H2 H2).x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:x < y
Hyp:0 < 2
H4:- (PI / 2) < (x + y) / 2
H5:x + y < y + y
y + y <= PI / 2 + PI / 2 -> sin x < sin y
  rewrite <- double_var.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:x < y
Hyp:0 < 2
H4:- (PI / 2) < (x + y) / 2
H5:x + y < y + y
y + y <= PI -> sin x < sin y
  intro H6; generalize (Rlt_le_trans (x + y) (y + y) PI H5 H6); intro H7;
    generalize (Rmult_lt_compat_l (/ 2) (x + y) PI (Rinv_0_lt_compat 2 Hyp) H7);
      replace (/ 2 * PI) with (PI / 2) by apply Rmult_comm.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:x < y
Hyp:0 < 2
H4:- (PI / 2) < (x + y) / 2
H5:x + y < y + y
H6:y + y <= PI
H7:x + y < PI
/ 2 * (x + y) < PI / 2 -> sin x < sin y
  replace (/ 2 * (x + y)) with ((x + y) / 2) by apply Rmult_comm.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H1:- (PI / 2) <= y
H2:y <= PI / 2
H3:x < y
Hyp:0 < 2
H4:- (PI / 2) < (x + y) / 2
H5:x + y < y + y
H6:y + y <= PI
H7:x + y < PI
(x + y) / 2 < PI / 2 -> sin x < sin y
  clear H5 H6 H7; intro H5; generalize (Ropp_le_ge_contravar (- (PI / 2)) y H1);
    rewrite Ropp_involutive; clear H1; intro H1;
      generalize (Rge_le (PI / 2) (- y) H1); clear H1; intro H1;
        generalize (Ropp_le_ge_contravar y (PI / 2) H2); clear H2;
          intro H2; generalize (Rge_le (- y) (- (PI / 2)) H2);
            clear H2; intro H2; generalize (Rplus_lt_compat_l (- y) x y H3);
              replace (- y + x) with (x - y) by apply Rplus_comm.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H3:x < y
Hyp:0 < 2
H4:- (PI / 2) < (x + y) / 2
H5:(x + y) / 2 < PI / 2
H1:- y <= PI / 2
H2:- (PI / 2) <= - y
x - y < - y + y -> sin x < sin y
  rewrite Rplus_opp_l.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H3:x < y
Hyp:0 < 2
H4:- (PI / 2) < (x + y) / 2
H5:(x + y) / 2 < PI / 2
H1:- y <= PI / 2
H2:- (PI / 2) <= - y
x - y < 0 -> sin x < sin y
  intro H6;
    generalize (Rmult_lt_compat_l (/ 2) (x - y) 0 (Rinv_0_lt_compat 2 Hyp) H6);
      rewrite Rmult_0_r; replace (/ 2 * (x - y)) with ((x - y) / 2) by apply Rmult_comm.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H3:x < y
Hyp:0 < 2
H4:- (PI / 2) < (x + y) / 2
H5:(x + y) / 2 < PI / 2
H1:- y <= PI / 2
H2:- (PI / 2) <= - y
H6:x - y < 0
(x - y) / 2 < 0 -> sin x < sin y
  clear H6; intro H6;
    generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) (- y) H H2);
      replace (- (PI / 2) + - (PI / 2)) with (- PI) by field.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H3:x < y
Hyp:0 < 2
H4:- (PI / 2) < (x + y) / 2
H5:(x + y) / 2 < PI / 2
H1:- y <= PI / 2
H2:- (PI / 2) <= - y
H6:(x - y) / 2 < 0
- PI <= x + - y -> sin x < sin y
  intro H7;
    generalize
      (Rmult_le_compat_l (/ 2) (- PI) (x - y)
        (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H7);
      replace (/ 2 * - PI) with (- (PI / 2)) by field.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H3:x < y
Hyp:0 < 2
H4:- (PI / 2) < (x + y) / 2
H5:(x + y) / 2 < PI / 2
H1:- y <= PI / 2
H2:- (PI / 2) <= - y
H6:(x - y) / 2 < 0
H7:- PI <= x + - y
- (PI / 2) <= / 2 * (x - y) -> sin x < sin y
  replace (/ 2 * (x - y)) with ((x - y) / 2) by apply Rmult_comm.x, y:R
H:- (PI / 2) <= x
H0:x <= PI / 2
H3:x < y
Hyp:0 < 2
H4:- (PI / 2) < (x + y) / 2
H5:(x + y) / 2 < PI / 2
H1:- y <= PI / 2
H2:- (PI / 2) <= - y
H6:(x - y) / 2 < 0
H7:- PI <= x + - y
- (PI / 2) <= (x - y) / 2 -> sin x < sin y
  clear H7; intro H7; clear H H0 H1 H2; apply Rminus_lt; rewrite form4;
    generalize (cos_gt_0 ((x + y) / 2) H4 H5); intro H8;
      generalize (Rmult_lt_0_compat 2 (cos ((x + y) / 2)) Hyp H8);
        clear H8; intro H8; cut (- PI < - (PI / 2)).x, y:R
H3:x < y
Hyp:0 < 2
H4:- (PI / 2) < (x + y) / 2
H5:(x + y) / 2 < PI / 2
H6:(x - y) / 2 < 0
H7:- (PI / 2) <= (x - y) / 2
H8:0 < 2 * cos ((x + y) / 2)
- PI < - (PI / 2) ->
2 * cos ((x + y) / 2) * sin ((x - y) / 2) < 0
x, y:R
H3:x < y
Hyp:0 < 2
H4:- (PI / 2) < (x + y) / 2
H5:(x + y) / 2 < PI / 2
H6:(x - y) / 2 < 0
H7:- (PI / 2) <= (x - y) / 2
H8:0 < 2 * cos ((x + y) / 2)
- PI < - (PI / 2)
  intro H9;
    generalize
      (sin_lt_0_var ((x - y) / 2)
        (Rlt_le_trans (- PI) (- (PI / 2)) ((x - y) / 2) H9 H7) H6);
      intro H10;
        generalize
          (Rmult_lt_gt_compat_neg_l (sin ((x - y) / 2)) 0 (
            2 * cos ((x + y) / 2)) H10 H8); intro H11; rewrite Rmult_0_r in H11;
          rewrite Rmult_comm; assumption.x, y:R
H3:x < y
Hyp:0 < 2
H4:- (PI / 2) < (x + y) / 2
H5:(x + y) / 2 < PI / 2
H6:(x - y) / 2 < 0
H7:- (PI / 2) <= (x - y) / 2
H8:0 < 2 * cos ((x + y) / 2)
- PI < - (PI / 2)
  apply Ropp_lt_gt_contravar; apply PI2_Rlt_PI.
Qed.

Lemma sin_decreasing_0 :
  forall x y:R,
    x <= 3 * (PI / 2) ->
    PI / 2 <= x -> y <= 3 * (PI / 2) -> PI / 2 <= y -> sin x < sin y -> y < x.forall x y : R,
x <= 3 * (PI / 2) ->
PI / 2 <= x ->
y <= 3 * (PI / 2) ->
PI / 2 <= y -> sin x < sin y -> y < x
Proof.forall x y : R,
x <= 3 * (PI / 2) ->
PI / 2 <= x ->
y <= 3 * (PI / 2) ->
PI / 2 <= y -> sin x < sin y -> y < x
  intros; rewrite <- (sin_PI_x x) in H3; rewrite <- (sin_PI_x y) in H3;
    generalize (Ropp_lt_gt_contravar (sin (PI - x)) (sin (PI - y)) H3);
      repeat rewrite <- sin_neg;
        generalize (Rplus_le_compat_l (- PI) x (3 * (PI / 2)) H);
          generalize (Rplus_le_compat_l (- PI) (PI / 2) x H0);
            generalize (Rplus_le_compat_l (- PI) y (3 * (PI / 2)) H1);
              generalize (Rplus_le_compat_l (- PI) (PI / 2) y H2);
                replace (- PI + x) with (x - PI) by apply Rplus_comm.x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:sin (PI - x) < sin (PI - y)
- PI + PI / 2 <= - PI + y ->
- PI + y <= - PI + 3 * (PI / 2) ->
- PI + PI / 2 <= x - PI ->
x - PI <= - PI + 3 * (PI / 2) ->
sin (- (PI - x)) > sin (- (PI - y)) -> y < x
  replace (- PI + PI / 2) with (- (PI / 2)) by field.x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:sin (PI - x) < sin (PI - y)
- (PI / 2) <= - PI + y ->
- PI + y <= - PI + 3 * (PI / 2) ->
- (PI / 2) <= x - PI ->
x - PI <= - PI + 3 * (PI / 2) ->
sin (- (PI - x)) > sin (- (PI - y)) -> y < x
  replace (- PI + y) with (y - PI) by apply Rplus_comm.x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:sin (PI - x) < sin (PI - y)
- (PI / 2) <= y - PI ->
y - PI <= - PI + 3 * (PI / 2) ->
- (PI / 2) <= x - PI ->
x - PI <= - PI + 3 * (PI / 2) ->
sin (- (PI - x)) > sin (- (PI - y)) -> y < x
  replace (- PI + 3 * (PI / 2)) with (PI / 2) by field.x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:sin (PI - x) < sin (PI - y)
- (PI / 2) <= y - PI ->
y - PI <= PI / 2 ->
- (PI / 2) <= x - PI ->
x - PI <= PI / 2 ->
sin (- (PI - x)) > sin (- (PI - y)) -> y < x
  replace (- (PI - x)) with (x - PI) by ring.x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:sin (PI - x) < sin (PI - y)
- (PI / 2) <= y - PI ->
y - PI <= PI / 2 ->
- (PI / 2) <= x - PI ->
x - PI <= PI / 2 ->
sin (x - PI) > sin (- (PI - y)) -> y < x
  replace (- (PI - y)) with (y - PI) by ring.x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:sin (PI - x) < sin (PI - y)
- (PI / 2) <= y - PI ->
y - PI <= PI / 2 ->
- (PI / 2) <= x - PI ->
x - PI <= PI / 2 ->
sin (x - PI) > sin (y - PI) -> y < x
  intros; change (sin (y - PI) < sin (x - PI)) in H8;
    apply Rplus_lt_reg_l with (- PI); rewrite Rplus_comm.x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:sin (PI - x) < sin (PI - y)
H4:- (PI / 2) <= y - PI
H5:y - PI <= PI / 2
H6:- (PI / 2) <= x - PI
H7:x - PI <= PI / 2
H8:sin (y - PI) < sin (x - PI)
y + - PI < - PI + x
  rewrite (Rplus_comm _ x).x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:sin (PI - x) < sin (PI - y)
H4:- (PI / 2) <= y - PI
H5:y - PI <= PI / 2
H6:- (PI / 2) <= x - PI
H7:x - PI <= PI / 2
H8:sin (y - PI) < sin (x - PI)
y + - PI < x + - PI
  apply (sin_increasing_0 (y - PI) (x - PI) H4 H5 H6 H7 H8).
Qed.

Lemma sin_decreasing_1 :
  forall x y:R,
    x <= 3 * (PI / 2) ->
    PI / 2 <= x -> y <= 3 * (PI / 2) -> PI / 2 <= y -> x < y -> sin y < sin x.forall x y : R,
x <= 3 * (PI / 2) ->
PI / 2 <= x ->
y <= 3 * (PI / 2) ->
PI / 2 <= y -> x < y -> sin y < sin x
Proof.forall x y : R,
x <= 3 * (PI / 2) ->
PI / 2 <= x ->
y <= 3 * (PI / 2) ->
PI / 2 <= y -> x < y -> sin y < sin x
  intros; rewrite <- (sin_PI_x x); rewrite <- (sin_PI_x y);
    generalize (Rplus_le_compat_l (- PI) x (3 * (PI / 2)) H);
      generalize (Rplus_le_compat_l (- PI) (PI / 2) x H0);
        generalize (Rplus_le_compat_l (- PI) y (3 * (PI / 2)) H1);
          generalize (Rplus_le_compat_l (- PI) (PI / 2) y H2);
            generalize (Rplus_lt_compat_l (- PI) x y H3);
              replace (- PI + PI / 2) with (- (PI / 2)) by field.x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:x < y
- PI + x < - PI + y ->
- (PI / 2) <= - PI + y ->
- PI + y <= - PI + 3 * (PI / 2) ->
- (PI / 2) <= - PI + x ->
- PI + x <= - PI + 3 * (PI / 2) ->
sin (PI - y) < sin (PI - x)
  replace (- PI + y) with (y - PI) by apply Rplus_comm.x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:x < y
- PI + x < y - PI ->
- (PI / 2) <= y - PI ->
y - PI <= - PI + 3 * (PI / 2) ->
- (PI / 2) <= - PI + x ->
- PI + x <= - PI + 3 * (PI / 2) ->
sin (PI - y) < sin (PI - x)
  replace (- PI + 3 * (PI / 2)) with (PI / 2) by field.x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:x < y
- PI + x < y - PI ->
- (PI / 2) <= y - PI ->
y - PI <= PI / 2 ->
- (PI / 2) <= - PI + x ->
- PI + x <= PI / 2 -> sin (PI - y) < sin (PI - x)
  replace (- PI + x) with (x - PI) by apply Rplus_comm.x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:x < y
x - PI < y - PI ->
- (PI / 2) <= y - PI ->
y - PI <= PI / 2 ->
- (PI / 2) <= x - PI ->
x - PI <= PI / 2 -> sin (PI - y) < sin (PI - x)
  intros; apply Ropp_lt_cancel; repeat rewrite <- sin_neg;
    replace (- (PI - x)) with (x - PI) by ring.x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:x < y
H4:x - PI < y - PI
H5:- (PI / 2) <= y - PI
H6:y - PI <= PI / 2
H7:- (PI / 2) <= x - PI
H8:x - PI <= PI / 2
sin (x - PI) < sin (- (PI - y))
  replace (- (PI - y)) with (y - PI) by ring.x, y:R
H:x <= 3 * (PI / 2)
H0:PI / 2 <= x
H1:y <= 3 * (PI / 2)
H2:PI / 2 <= y
H3:x < y
H4:x - PI < y - PI
H5:- (PI / 2) <= y - PI
H6:y - PI <= PI / 2
H7:- (PI / 2) <= x - PI
H8:x - PI <= PI / 2
sin (x - PI) < sin (y - PI)
  apply (sin_increasing_1 (x - PI) (y - PI) H7 H8 H5 H6 H4).
Qed.

Lemma cos_increasing_0 :
  forall x y:R,
    PI <= x -> x <= 2 * PI -> PI <= y -> y <= 2 * PI -> cos x < cos y -> x < y.forall x y : R,
PI <= x ->
x <= 2 * PI ->
PI <= y -> y <= 2 * PI -> cos x < cos y -> x < y
Proof.forall x y : R,
PI <= x ->
x <= 2 * PI ->
PI <= y -> y <= 2 * PI -> cos x < cos y -> x < y
  intros x y H1 H2 H3 H4; rewrite <- (cos_neg x); rewrite <- (cos_neg y);
    rewrite <- (cos_period (- x) 1); rewrite <- (cos_period (- y) 1);
      unfold INR in |- *;
        replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))) by field.x, y:R
H1:PI <= x
H2:x <= 2 * PI
H3:PI <= y
H4:y <= 2 * PI
cos (PI / 2 - (x - 3 * (PI / 2))) <
cos (- y + 2 * 1 * PI) -> x < y
  replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))) by field.x, y:R
H1:PI <= x
H2:x <= 2 * PI
H3:PI <= y
H4:y <= 2 * PI
cos (PI / 2 - (x - 3 * (PI / 2))) <
cos (PI / 2 - (y - 3 * (PI / 2))) -> x < y
  repeat rewrite cos_shift; intro H5;
    generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI x H1);
      generalize (Rplus_le_compat_l (-3 * (PI / 2)) x (2 * PI) H2);
        generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI y H3);
          generalize (Rplus_le_compat_l (-3 * (PI / 2)) y (2 * PI) H4).x, y:R
H1:PI <= x
H2:x <= 2 * PI
H3:PI <= y
H4:y <= 2 * PI
H5:sin (x - 3 * (PI / 2)) < sin (y - 3 * (PI / 2))
-3 * (PI / 2) + y <= -3 * (PI / 2) + 2 * PI ->
-3 * (PI / 2) + PI <= -3 * (PI / 2) + y ->
-3 * (PI / 2) + x <= -3 * (PI / 2) + 2 * PI ->
-3 * (PI / 2) + PI <= -3 * (PI / 2) + x -> x < y
  replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)) by ring.x, y:R
H1:PI <= x
H2:x <= 2 * PI
H3:PI <= y
H4:y <= 2 * PI
H5:sin (x - 3 * (PI / 2)) < sin (y - 3 * (PI / 2))
y - 3 * (PI / 2) <= -3 * (PI / 2) + 2 * PI ->
-3 * (PI / 2) + PI <= y - 3 * (PI / 2) ->
-3 * (PI / 2) + x <= -3 * (PI / 2) + 2 * PI ->
-3 * (PI / 2) + PI <= -3 * (PI / 2) + x -> x < y
  replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)) by ring.x, y:R
H1:PI <= x
H2:x <= 2 * PI
H3:PI <= y
H4:y <= 2 * PI
H5:sin (x - 3 * (PI / 2)) < sin (y - 3 * (PI / 2))
y - 3 * (PI / 2) <= -3 * (PI / 2) + 2 * PI ->
-3 * (PI / 2) + PI <= y - 3 * (PI / 2) ->
x - 3 * (PI / 2) <= -3 * (PI / 2) + 2 * PI ->
-3 * (PI / 2) + PI <= x - 3 * (PI / 2) -> x < y
  replace (-3 * (PI / 2) + 2 * PI) with (PI / 2) by field.x, y:R
H1:PI <= x
H2:x <= 2 * PI
H3:PI <= y
H4:y <= 2 * PI
H5:sin (x - 3 * (PI / 2)) < sin (y - 3 * (PI / 2))
y - 3 * (PI / 2) <= PI / 2 ->
-3 * (PI / 2) + PI <= y - 3 * (PI / 2) ->
x - 3 * (PI / 2) <= PI / 2 ->
-3 * (PI / 2) + PI <= x - 3 * (PI / 2) -> x < y
  replace (-3 * (PI / 2) + PI) with (- (PI / 2)) by field.x, y:R
H1:PI <= x
H2:x <= 2 * PI
H3:PI <= y
H4:y <= 2 * PI
H5:sin (x - 3 * (PI / 2)) < sin (y - 3 * (PI / 2))
y - 3 * (PI / 2) <= PI / 2 ->
- (PI / 2) <= y - 3 * (PI / 2) ->
x - 3 * (PI / 2) <= PI / 2 ->
- (PI / 2) <= x - 3 * (PI / 2) -> x < y
  clear H1 H2 H3 H4; intros H1 H2 H3 H4;
    apply Rplus_lt_reg_l with (-3 * (PI / 2));
      replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)) by ring.x, y:R
H5:sin (x - 3 * (PI / 2)) < sin (y - 3 * (PI / 2))
H1:y - 3 * (PI / 2) <= PI / 2
H2:- (PI / 2) <= y - 3 * (PI / 2)
H3:x - 3 * (PI / 2) <= PI / 2
H4:- (PI / 2) <= x - 3 * (PI / 2)
x - 3 * (PI / 2) < -3 * (PI / 2) + y
  replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)) by ring.x, y:R
H5:sin (x - 3 * (PI / 2)) < sin (y - 3 * (PI / 2))
H1:y - 3 * (PI / 2) <= PI / 2
H2:- (PI / 2) <= y - 3 * (PI / 2)
H3:x - 3 * (PI / 2) <= PI / 2
H4:- (PI / 2) <= x - 3 * (PI / 2)
x - 3 * (PI / 2) < y - 3 * (PI / 2)
  apply (sin_increasing_0 (x - 3 * (PI / 2)) (y - 3 * (PI / 2)) H4 H3 H2 H1 H5).
Qed.

Lemma cos_increasing_1 :
  forall x y:R,
    PI <= x -> x <= 2 * PI -> PI <= y -> y <= 2 * PI -> x < y -> cos x < cos y.forall x y : R,
PI <= x ->
x <= 2 * PI ->
PI <= y -> y <= 2 * PI -> x < y -> cos x < cos y
Proof.forall x y : R,
PI <= x ->
x <= 2 * PI ->
PI <= y -> y <= 2 * PI -> x < y -> cos x < cos y
  intros x y H1 H2 H3 H4 H5;
    generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI x H1);
      generalize (Rplus_le_compat_l (-3 * (PI / 2)) x (2 * PI) H2);
        generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI y H3);
          generalize (Rplus_le_compat_l (-3 * (PI / 2)) y (2 * PI) H4);
            generalize (Rplus_lt_compat_l (-3 * (PI / 2)) x y H5);
              rewrite <- (cos_neg x); rewrite <- (cos_neg y);
                rewrite <- (cos_period (- x) 1); rewrite <- (cos_period (- y) 1);
                  unfold INR in |- *; replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)) by ring.x, y:R
H1:PI <= x
H2:x <= 2 * PI
H3:PI <= y
H4:y <= 2 * PI
H5:x < y
x - 3 * (PI / 2) < -3 * (PI / 2) + y ->
-3 * (PI / 2) + y <= -3 * (PI / 2) + 2 * PI ->
-3 * (PI / 2) + PI <= -3 * (PI / 2) + y ->
x - 3 * (PI / 2) <= -3 * (PI / 2) + 2 * PI ->
-3 * (PI / 2) + PI <= x - 3 * (PI / 2) ->
cos (- x + 2 * 1 * PI) < cos (- y + 2 * 1 * PI)
  replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)) by ring.x, y:R
H1:PI <= x
H2:x <= 2 * PI
H3:PI <= y
H4:y <= 2 * PI
H5:x < y
x - 3 * (PI / 2) < y - 3 * (PI / 2) ->
y - 3 * (PI / 2) <= -3 * (PI / 2) + 2 * PI ->
-3 * (PI / 2) + PI <= y - 3 * (PI / 2) ->
x - 3 * (PI / 2) <= -3 * (PI / 2) + 2 * PI ->
-3 * (PI / 2) + PI <= x - 3 * (PI / 2) ->
cos (- x + 2 * 1 * PI) < cos (- y + 2 * 1 * PI)
  replace (-3 * (PI / 2) + PI) with (- (PI / 2)) by field.x, y:R
H1:PI <= x
H2:x <= 2 * PI
H3:PI <= y
H4:y <= 2 * PI
H5:x < y
x - 3 * (PI / 2) < y - 3 * (PI / 2) ->
y - 3 * (PI / 2) <= -3 * (PI / 2) + 2 * PI ->
- (PI / 2) <= y - 3 * (PI / 2) ->
x - 3 * (PI / 2) <= -3 * (PI / 2) + 2 * PI ->
- (PI / 2) <= x - 3 * (PI / 2) ->
cos (- x + 2 * 1 * PI) < cos (- y + 2 * 1 * PI)
  replace (-3 * (PI / 2) + 2 * PI) with (PI / 2) by field.x, y:R
H1:PI <= x
H2:x <= 2 * PI
H3:PI <= y
H4:y <= 2 * PI
H5:x < y
x - 3 * (PI / 2) < y - 3 * (PI / 2) ->
y - 3 * (PI / 2) <= PI / 2 ->
- (PI / 2) <= y - 3 * (PI / 2) ->
x - 3 * (PI / 2) <= PI / 2 ->
- (PI / 2) <= x - 3 * (PI / 2) ->
cos (- x + 2 * 1 * PI) < cos (- y + 2 * 1 * PI)
  clear H1 H2 H3 H4 H5; intros H1 H2 H3 H4 H5;
    replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))) by field.x, y:R
H1:x - 3 * (PI / 2) < y - 3 * (PI / 2)
H2:y - 3 * (PI / 2) <= PI / 2
H3:- (PI / 2) <= y - 3 * (PI / 2)
H4:x - 3 * (PI / 2) <= PI / 2
H5:- (PI / 2) <= x - 3 * (PI / 2)
cos (PI / 2 - (x - 3 * (PI / 2))) <
cos (- y + 2 * 1 * PI)
  replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))) by field.x, y:R
H1:x - 3 * (PI / 2) < y - 3 * (PI / 2)
H2:y - 3 * (PI / 2) <= PI / 2
H3:- (PI / 2) <= y - 3 * (PI / 2)
H4:x - 3 * (PI / 2) <= PI / 2
H5:- (PI / 2) <= x - 3 * (PI / 2)
cos (PI / 2 - (x - 3 * (PI / 2))) <
cos (PI / 2 - (y - 3 * (PI / 2)))
  repeat rewrite cos_shift;
    apply
      (sin_increasing_1 (x - 3 * (PI / 2)) (y - 3 * (PI / 2)) H5 H4 H3 H2 H1).
Qed.

Lemma cos_decreasing_0 :
  forall x y:R,
    0 <= x -> x <= PI -> 0 <= y -> y <= PI -> cos x < cos y -> y < x.forall x y : R,
0 <= x ->
x <= PI -> 0 <= y -> y <= PI -> cos x < cos y -> y < x
Proof.forall x y : R,
0 <= x ->
x <= PI -> 0 <= y -> y <= PI -> cos x < cos y -> y < x
  intros; generalize (Ropp_lt_gt_contravar (cos x) (cos y) H3);
    repeat rewrite <- neg_cos; intro H4;
      change (cos (y + PI) < cos (x + PI)) in H4; rewrite (Rplus_comm x) in H4;
        rewrite (Rplus_comm y) in H4; generalize (Rplus_le_compat_l PI 0 x H);
          generalize (Rplus_le_compat_l PI x PI H0);
            generalize (Rplus_le_compat_l PI 0 y H1);
              generalize (Rplus_le_compat_l PI y PI H2); rewrite Rplus_0_r.x, y:R
H:0 <= x
H0:x <= PI
H1:0 <= y
H2:y <= PI
H3:cos x < cos y
H4:cos (PI + y) < cos (PI + x)
PI + y <= PI + PI ->
PI <= PI + y ->
PI + x <= PI + PI -> PI <= PI + x -> y < x
  rewrite <- double.x, y:R
H:0 <= x
H0:x <= PI
H1:0 <= y
H2:y <= PI
H3:cos x < cos y
H4:cos (PI + y) < cos (PI + x)
PI + y <= 2 * PI ->
PI <= PI + y ->
PI + x <= 2 * PI -> PI <= PI + x -> y < x
  clear H H0 H1 H2 H3; intros; apply Rplus_lt_reg_l with PI;
    apply (cos_increasing_0 (PI + y) (PI + x) H0 H H2 H1 H4).
Qed.

Lemma cos_decreasing_1 :
  forall x y:R,
    0 <= x -> x <= PI -> 0 <= y -> y <= PI -> x < y -> cos y < cos x.forall x y : R,
0 <= x ->
x <= PI -> 0 <= y -> y <= PI -> x < y -> cos y < cos x
Proof.forall x y : R,
0 <= x ->
x <= PI -> 0 <= y -> y <= PI -> x < y -> cos y < cos x
  intros; apply Ropp_lt_cancel; repeat rewrite <- neg_cos;
    rewrite (Rplus_comm x); rewrite (Rplus_comm y);
      generalize (Rplus_le_compat_l PI 0 x H);
        generalize (Rplus_le_compat_l PI x PI H0);
          generalize (Rplus_le_compat_l PI 0 y H1);
            generalize (Rplus_le_compat_l PI y PI H2); rewrite Rplus_0_r.x, y:R
H:0 <= x
H0:x <= PI
H1:0 <= y
H2:y <= PI
H3:x < y
PI + y <= PI + PI ->
PI <= PI + y ->
PI + x <= PI + PI ->
PI <= PI + x -> cos (PI + x) < cos (PI + y)
  rewrite <- double.x, y:R
H:0 <= x
H0:x <= PI
H1:0 <= y
H2:y <= PI
H3:x < y
PI + y <= 2 * PI ->
PI <= PI + y ->
PI + x <= 2 * PI ->
PI <= PI + x -> cos (PI + x) < cos (PI + y)
  generalize (Rplus_lt_compat_l PI x y H3); clear H H0 H1 H2 H3; intros;
    apply (cos_increasing_1 (PI + x) (PI + y) H3 H2 H1 H0 H).
Qed.

Lemma tan_diff :
  forall x y:R,
    cos x <> 0 -> cos y <> 0 -> tan x - tan y = sin (x - y) / (cos x * cos y).forall x y : R,
cos x <> 0 ->
cos y <> 0 ->
tan x - tan y = sin (x - y) / (cos x * cos y)
Proof.forall x y : R,
cos x <> 0 ->
cos y <> 0 ->
tan x - tan y = sin (x - y) / (cos x * cos y)
  intros; unfold tan in |- *; rewrite sin_minus.x, y:R
H:cos x <> 0
H0:cos y <> 0
sin x / cos x - sin y / cos y =
(sin x * cos y - cos x * sin y) / (cos x * cos y)
  field.x, y:R
H:cos x <> 0
H0:cos y <> 0
cos y <> 0 /\ cos x <> 0
  now split.
Qed.

Lemma tan_increasing_0 :
  forall x y:R,
    - (PI / 4) <= x ->
    x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> tan x < tan y -> x < y.forall x y : R,
- (PI / 4) <= x ->
x <= PI / 4 ->
- (PI / 4) <= y ->
y <= PI / 4 -> tan x < tan y -> x < y
Proof.forall x y : R,
- (PI / 4) <= x ->
x <= PI / 4 ->
- (PI / 4) <= y ->
y <= PI / 4 -> tan x < tan y -> x < y
  intros; generalize PI4_RLT_PI2; intro H4;
    generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4);
      intro H5; change (- (PI / 2) < - (PI / 4)) in H5;
        generalize
          (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
            (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)); intro HP1;
          generalize
            (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
              (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)); intro HP2;
            generalize
              (not_eq_sym
                (Rlt_not_eq 0 (cos x)
                  (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
                    (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4))));
              intro H6;
                generalize
                  (not_eq_sym
                    (Rlt_not_eq 0 (cos y)
                      (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
                        (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4))));
                  intro H7; generalize (tan_diff x y H6 H7); intro H8;
                    generalize (Rlt_minus (tan x) (tan y) H3); clear H3;
                      intro H3; rewrite H8 in H3; cut (sin (x - y) < 0).x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
sin (x - y) < 0 -> x < y
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
sin (x - y) < 0
  intro H9; generalize (Ropp_le_ge_contravar (- (PI / 4)) y H1);
    rewrite Ropp_involutive; intro H10; generalize (Rge_le (PI / 4) (- y) H10);
      clear H10; intro H10; generalize (Ropp_le_ge_contravar y (PI / 4) H2);
        intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11);
          clear H11; intro H11;
            generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11);
              generalize (Rplus_le_compat x (PI / 4) (- y) (PI / 4) H0 H10).x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:sin (x - y) < 0
H10:- y <= PI / 4
H11:- (PI / 4) <= - y
x + - y <= PI / 4 + PI / 4 ->
- (PI / 4) + - (PI / 4) <= x + - y -> x < y
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
sin (x - y) < 0
  replace (PI / 4 + PI / 4) with (PI / 2) by field.x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:sin (x - y) < 0
H10:- y <= PI / 4
H11:- (PI / 4) <= - y
x + - y <= PI / 2 ->
- (PI / 4) + - (PI / 4) <= x + - y -> x < y
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
sin (x - y) < 0
  replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)) by field.x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:sin (x - y) < 0
H10:- y <= PI / 4
H11:- (PI / 4) <= - y
x + - y <= PI / 2 -> - (PI / 2) <= x + - y -> x < y
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
sin (x - y) < 0
  intros; case (Rtotal_order 0 (x - y)); intro H14.x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:sin (x - y) < 0
H10:- y <= PI / 4
H11:- (PI / 4) <= - y
H12:x + - y <= PI / 2
H13:- (PI / 2) <= x + - y
H14:0 < x - y
x < y
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:sin (x - y) < 0
H10:- y <= PI / 4
H11:- (PI / 4) <= - y
H12:x + - y <= PI / 2
H13:- (PI / 2) <= x + - y
H14:0 = x - y \/ 0 > x - y
x < y
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
sin (x - y) < 0
  generalize
    (sin_gt_0 (x - y) H14 (Rle_lt_trans (x - y) (PI / 2) PI H12 PI2_Rlt_PI));
    intro H15; elim (Rlt_irrefl 0 (Rlt_trans 0 (sin (x - y)) 0 H15 H9)).x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:sin (x - y) < 0
H10:- y <= PI / 4
H11:- (PI / 4) <= - y
H12:x + - y <= PI / 2
H13:- (PI / 2) <= x + - y
H14:0 = x - y \/ 0 > x - y
x < y
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
sin (x - y) < 0
  elim H14; intro H15.x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:sin (x - y) < 0
H10:- y <= PI / 4
H11:- (PI / 4) <= - y
H12:x + - y <= PI / 2
H13:- (PI / 2) <= x + - y
H14:0 = x - y \/ 0 > x - y
H15:0 = x - y
x < y
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:sin (x - y) < 0
H10:- y <= PI / 4
H11:- (PI / 4) <= - y
H12:x + - y <= PI / 2
H13:- (PI / 2) <= x + - y
H14:0 = x - y \/ 0 > x - y
H15:0 > x - y
x < y
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
sin (x - y) < 0
  rewrite <- H15 in H9; rewrite sin_0 in H9; elim (Rlt_irrefl 0 H9).x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:sin (x - y) < 0
H10:- y <= PI / 4
H11:- (PI / 4) <= - y
H12:x + - y <= PI / 2
H13:- (PI / 2) <= x + - y
H14:0 = x - y \/ 0 > x - y
H15:0 > x - y
x < y
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
sin (x - y) < 0
  apply Rminus_lt; assumption.x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
sin (x - y) < 0
  case (Rcase_abs (sin (x - y))); intro H9.x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:sin (x - y) < 0
sin (x - y) < 0
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:sin (x - y) >= 0
sin (x - y) < 0
  assumption.x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:sin (x - y) >= 0
sin (x - y) < 0
  generalize (Rge_le (sin (x - y)) 0 H9); clear H9; intro H9;
    generalize (Rinv_0_lt_compat (cos x) HP1); intro H10;
      generalize (Rinv_0_lt_compat (cos y) HP2); intro H11;
        generalize (Rmult_lt_0_compat (/ cos x) (/ cos y) H10 H11);
          replace (/ cos x * / cos y) with (/ (cos x * cos y)).x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:0 <= sin (x - y)
H10:0 < / cos x
H11:0 < / cos y
0 < / (cos x * cos y) -> sin (x - y) < 0
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:0 <= sin (x - y)
H10:0 < / cos x
H11:0 < / cos y
/ (cos x * cos y) = / cos x * / cos y
  intro H12;
    generalize
      (Rmult_le_pos (sin (x - y)) (/ (cos x * cos y)) H9
        (Rlt_le 0 (/ (cos x * cos y)) H12)); intro H13;
      elim
        (Rlt_irrefl 0 (Rle_lt_trans 0 (sin (x - y) * / (cos x * cos y)) 0 H13 H3)).x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:0 <= sin (x - y)
H10:0 < / cos x
H11:0 < / cos y
/ (cos x * cos y) = / cos x * / cos y
  apply Rinv_mult_distr.x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:0 <= sin (x - y)
H10:0 < / cos x
H11:0 < / cos y
cos x <> 0
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:0 <= sin (x - y)
H10:0 < / cos x
H11:0 < / cos y
cos y <> 0
  assumption.x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:tan x - tan y = sin (x - y) / (cos x * cos y)
H3:sin (x - y) / (cos x * cos y) < 0
H9:0 <= sin (x - y)
H10:0 < / cos x
H11:0 < / cos y
cos y <> 0
  assumption.
Qed.

Lemma tan_increasing_1 :
  forall x y:R,
    - (PI / 4) <= x ->
    x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> x < y -> tan x < tan y.forall x y : R,
- (PI / 4) <= x ->
x <= PI / 4 ->
- (PI / 4) <= y ->
y <= PI / 4 -> x < y -> tan x < tan y
Proof.forall x y : R,
- (PI / 4) <= x ->
x <= PI / 4 ->
- (PI / 4) <= y ->
y <= PI / 4 -> x < y -> tan x < tan y
  intros; apply Rminus_lt; generalize PI4_RLT_PI2; intro H4;
    generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4);
      intro H5; change (- (PI / 2) < - (PI / 4)) in H5;
        generalize
          (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
            (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)); intro HP1;
          generalize
            (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
              (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)); intro HP2;
            generalize
              (not_eq_sym
                (Rlt_not_eq 0 (cos x)
                  (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
                    (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4))));
              intro H6;
                generalize
                  (not_eq_sym
                    (Rlt_not_eq 0 (cos y)
                      (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
                        (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4))));
                  intro H7; rewrite (tan_diff x y H6 H7);
                    generalize (Rinv_0_lt_compat (cos x) HP1); intro H10;
                      generalize (Rinv_0_lt_compat (cos y) HP2); intro H11;
                        generalize (Rmult_lt_0_compat (/ cos x) (/ cos y) H10 H11);
                          replace (/ cos x * / cos y) with (/ (cos x * cos y)).x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H3:x < y
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H10:0 < / cos x
H11:0 < / cos y
0 < / (cos x * cos y) ->
sin (x - y) / (cos x * cos y) < 0
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H3:x < y
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H10:0 < / cos x
H11:0 < / cos y
/ (cos x * cos y) = / cos x * / cos y
  clear H10 H11; intro H8; generalize (Ropp_le_ge_contravar y (PI / 4) H2);
    intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11);
      clear H11; intro H11;
        generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11).x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H3:x < y
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:0 < / (cos x * cos y)
H11:- (PI / 4) <= - y
- (PI / 4) + - (PI / 4) <= x + - y ->
sin (x - y) / (cos x * cos y) < 0
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H3:x < y
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H10:0 < / cos x
H11:0 < / cos y
/ (cos x * cos y) = / cos x * / cos y
  replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)) by field.x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H3:x < y
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H8:0 < / (cos x * cos y)
H11:- (PI / 4) <= - y
- (PI / 2) <= x + - y ->
sin (x - y) / (cos x * cos y) < 0
x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H3:x < y
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H10:0 < / cos x
H11:0 < / cos y
/ (cos x * cos y) = / cos x * / cos y
  clear H11; intro H9; generalize (Rlt_minus x y H3); clear H3; intro H3;
    clear H H0 H1 H2 H4 H5 HP1 HP2; generalize PI2_Rlt_PI;
      intro H1; generalize (Ropp_lt_gt_contravar (PI / 2) PI H1);
        clear H1; intro H1;
          generalize
            (sin_lt_0_var (x - y) (Rlt_le_trans (- PI) (- (PI / 2)) (x - y) H1 H9) H3);
            intro H2;
              generalize
                (Rmult_lt_gt_compat_neg_l (sin (x - y)) 0 (/ (cos x * cos y)) H2 H8);
                rewrite Rmult_0_r; intro H4; assumption.x, y:R
H:- (PI / 4) <= x
H0:x <= PI / 4
H1:- (PI / 4) <= y
H2:y <= PI / 4
H3:x < y
H4:PI / 4 < PI / 2
H5:- (PI / 2) < - (PI / 4)
HP1:0 < cos x
HP2:0 < cos y
H6:cos x <> 0
H7:cos y <> 0
H10:0 < / cos x
H11:0 < / cos y
/ (cos x * cos y) = / cos x * / cos y
  apply Rinv_mult_distr; assumption.
Qed.

Lemma sin_incr_0 :
  forall x y:R,
    - (PI / 2) <= x ->
    x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> sin x <= sin y -> x <= y.forall x y : R,
- (PI / 2) <= x ->
x <= PI / 2 ->
- (PI / 2) <= y ->
y <= PI / 2 -> sin x <= sin y -> x <= y
Proof.forall x y : R,
- (PI / 2) <= x ->
x <= PI / 2 ->
- (PI / 2) <= y ->
y <= PI / 2 -> sin x <= sin y -> x <= y
  intros; case (Rtotal_order (sin x) (sin y)); intro H4;
    [ left; apply (sin_increasing_0 x y H H0 H1 H2 H4)
      | elim H4; intro H5;
        [ case (Rtotal_order x y); intro H6;
          [ left; assumption
            | elim H6; intro H7;
              [ right; assumption
                | generalize (sin_increasing_1 y x H1 H2 H H0 H7); intro H8;
                  rewrite H5 in H8; elim (Rlt_irrefl (sin y) H8) ] ]
          | elim (Rlt_irrefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5)) ] ].
Qed.

Lemma sin_incr_1 :
  forall x y:R,
    - (PI / 2) <= x ->
    x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> x <= y -> sin x <= sin y.forall x y : R,
- (PI / 2) <= x ->
x <= PI / 2 ->
- (PI / 2) <= y ->
y <= PI / 2 -> x <= y -> sin x <= sin y
Proof.forall x y : R,
- (PI / 2) <= x ->
x <= PI / 2 ->
- (PI / 2) <= y ->
y <= PI / 2 -> x <= y -> sin x <= sin y
  intros; case (Rtotal_order x y); intro H4;
    [ left; apply (sin_increasing_1 x y H H0 H1 H2 H4)
      | elim H4; intro H5;
        [ case (Rtotal_order (sin x) (sin y)); intro H6;
          [ left; assumption
            | elim H6; intro H7;
              [ right; assumption
                | generalize (sin_increasing_0 y x H1 H2 H H0 H7); intro H8;
                  rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ]
          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
Qed.

Lemma sin_decr_0 :
  forall x y:R,
    x <= 3 * (PI / 2) ->
    PI / 2 <= x ->
    y <= 3 * (PI / 2) -> PI / 2 <= y -> sin x <= sin y -> y <= x.forall x y : R,
x <= 3 * (PI / 2) ->
PI / 2 <= x ->
y <= 3 * (PI / 2) ->
PI / 2 <= y -> sin x <= sin y -> y <= x
Proof.forall x y : R,
x <= 3 * (PI / 2) ->
PI / 2 <= x ->
y <= 3 * (PI / 2) ->
PI / 2 <= y -> sin x <= sin y -> y <= x
  intros; case (Rtotal_order (sin x) (sin y)); intro H4;
    [ left; apply (sin_decreasing_0 x y H H0 H1 H2 H4)
      | elim H4; intro H5;
        [ case (Rtotal_order x y); intro H6;
          [ generalize (sin_decreasing_1 x y H H0 H1 H2 H6); intro H8;
            rewrite H5 in H8; elim (Rlt_irrefl (sin y) H8)
            | elim H6; intro H7;
              [ right; symmetry  in |- *; assumption | left; assumption ] ]
          | elim (Rlt_irrefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5)) ] ].
Qed.

Lemma sin_decr_1 :
  forall x y:R,
    x <= 3 * (PI / 2) ->
    PI / 2 <= x ->
    y <= 3 * (PI / 2) -> PI / 2 <= y -> x <= y -> sin y <= sin x.forall x y : R,
x <= 3 * (PI / 2) ->
PI / 2 <= x ->
y <= 3 * (PI / 2) ->
PI / 2 <= y -> x <= y -> sin y <= sin x
Proof.forall x y : R,
x <= 3 * (PI / 2) ->
PI / 2 <= x ->
y <= 3 * (PI / 2) ->
PI / 2 <= y -> x <= y -> sin y <= sin x
  intros; case (Rtotal_order x y); intro H4;
    [ left; apply (sin_decreasing_1 x y H H0 H1 H2 H4)
      | elim H4; intro H5;
        [ case (Rtotal_order (sin x) (sin y)); intro H6;
          [ generalize (sin_decreasing_0 x y H H0 H1 H2 H6); intro H8;
            rewrite H5 in H8; elim (Rlt_irrefl y H8)
            | elim H6; intro H7;
              [ right; symmetry  in |- *; assumption | left; assumption ] ]
          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
Qed.

Lemma cos_incr_0 :
  forall x y:R,
    PI <= x ->
    x <= 2 * PI -> PI <= y -> y <= 2 * PI -> cos x <= cos y -> x <= y.forall x y : R,
PI <= x ->
x <= 2 * PI ->
PI <= y -> y <= 2 * PI -> cos x <= cos y -> x <= y
Proof.forall x y : R,
PI <= x ->
x <= 2 * PI ->
PI <= y -> y <= 2 * PI -> cos x <= cos y -> x <= y
  intros; case (Rtotal_order (cos x) (cos y)); intro H4;
    [ left; apply (cos_increasing_0 x y H H0 H1 H2 H4)
      | elim H4; intro H5;
        [ case (Rtotal_order x y); intro H6;
          [ left; assumption
            | elim H6; intro H7;
              [ right; assumption
                | generalize (cos_increasing_1 y x H1 H2 H H0 H7); intro H8;
                  rewrite H5 in H8; elim (Rlt_irrefl (cos y) H8) ] ]
          | elim (Rlt_irrefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5)) ] ].
Qed.

Lemma cos_incr_1 :
  forall x y:R,
    PI <= x ->
    x <= 2 * PI -> PI <= y -> y <= 2 * PI -> x <= y -> cos x <= cos y.forall x y : R,
PI <= x ->
x <= 2 * PI ->
PI <= y -> y <= 2 * PI -> x <= y -> cos x <= cos y
Proof.forall x y : R,
PI <= x ->
x <= 2 * PI ->
PI <= y -> y <= 2 * PI -> x <= y -> cos x <= cos y
  intros; case (Rtotal_order x y); intro H4;
    [ left; apply (cos_increasing_1 x y H H0 H1 H2 H4)
      | elim H4; intro H5;
        [ case (Rtotal_order (cos x) (cos y)); intro H6;
          [ left; assumption
            | elim H6; intro H7;
              [ right; assumption
                | generalize (cos_increasing_0 y x H1 H2 H H0 H7); intro H8;
                  rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ]
          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
Qed.

Lemma cos_decr_0 :
  forall x y:R,
    0 <= x -> x <= PI -> 0 <= y -> y <= PI -> cos x <= cos y -> y <= x.forall x y : R,
0 <= x ->
x <= PI ->
0 <= y -> y <= PI -> cos x <= cos y -> y <= x
Proof.forall x y : R,
0 <= x ->
x <= PI ->
0 <= y -> y <= PI -> cos x <= cos y -> y <= x
  intros; case (Rtotal_order (cos x) (cos y)); intro H4;
    [ left; apply (cos_decreasing_0 x y H H0 H1 H2 H4)
      | elim H4; intro H5;
        [ case (Rtotal_order x y); intro H6;
          [ generalize (cos_decreasing_1 x y H H0 H1 H2 H6); intro H8;
            rewrite H5 in H8; elim (Rlt_irrefl (cos y) H8)
            | elim H6; intro H7;
              [ right; symmetry  in |- *; assumption | left; assumption ] ]
          | elim (Rlt_irrefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5)) ] ].
Qed.

Lemma cos_decr_1 :
  forall x y:R,
    0 <= x -> x <= PI -> 0 <= y -> y <= PI -> x <= y -> cos y <= cos x.forall x y : R,
0 <= x ->
x <= PI ->
0 <= y -> y <= PI -> x <= y -> cos y <= cos x
Proof.forall x y : R,
0 <= x ->
x <= PI ->
0 <= y -> y <= PI -> x <= y -> cos y <= cos x
  intros; case (Rtotal_order x y); intro H4;
    [ left; apply (cos_decreasing_1 x y H H0 H1 H2 H4)
      | elim H4; intro H5;
        [ case (Rtotal_order (cos x) (cos y)); intro H6;
          [ generalize (cos_decreasing_0 x y H H0 H1 H2 H6); intro H8;
            rewrite H5 in H8; elim (Rlt_irrefl y H8)
            | elim H6; intro H7;
              [ right; symmetry  in |- *; assumption | left; assumption ] ]
          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
Qed.

Lemma tan_incr_0 :
  forall x y:R,
    - (PI / 4) <= x ->
    x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> tan x <= tan y -> x <= y.forall x y : R,
- (PI / 4) <= x ->
x <= PI / 4 ->
- (PI / 4) <= y ->
y <= PI / 4 -> tan x <= tan y -> x <= y
Proof.forall x y : R,
- (PI / 4) <= x ->
x <= PI / 4 ->
- (PI / 4) <= y ->
y <= PI / 4 -> tan x <= tan y -> x <= y
  intros; case (Rtotal_order (tan x) (tan y)); intro H4;
    [ left; apply (tan_increasing_0 x y H H0 H1 H2 H4)
      | elim H4; intro H5;
        [ case (Rtotal_order x y); intro H6;
          [ left; assumption
            | elim H6; intro H7;
              [ right; assumption
                | generalize (tan_increasing_1 y x H1 H2 H H0 H7); intro H8;
                  rewrite H5 in H8; elim (Rlt_irrefl (tan y) H8) ] ]
          | elim (Rlt_irrefl (tan x) (Rle_lt_trans (tan x) (tan y) (tan x) H3 H5)) ] ].
Qed.

Lemma tan_incr_1 :
  forall x y:R,
    - (PI / 4) <= x ->
    x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> x <= y -> tan x <= tan y.forall x y : R,
- (PI / 4) <= x ->
x <= PI / 4 ->
- (PI / 4) <= y ->
y <= PI / 4 -> x <= y -> tan x <= tan y
Proof.forall x y : R,
- (PI / 4) <= x ->
x <= PI / 4 ->
- (PI / 4) <= y ->
y <= PI / 4 -> x <= y -> tan x <= tan y
  intros; case (Rtotal_order x y); intro H4;
    [ left; apply (tan_increasing_1 x y H H0 H1 H2 H4)
      | elim H4; intro H5;
        [ case (Rtotal_order (tan x) (tan y)); intro H6;
          [ left; assumption
            | elim H6; intro H7;
              [ right; assumption
                | generalize (tan_increasing_0 y x H1 H2 H H0 H7); intro H8;
                  rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ]
          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
Qed.

(**********)
Lemma sin_eq_0_1 : forall x:R, (exists k : Z, x = IZR k * PI) -> sin x = 0.forall x : R,
(exists k : Z, x = IZR k * PI) -> sin x = 0
Proof.forall x : R,
(exists k : Z, x = IZR k * PI) -> sin x = 0
  intros.x:R
H:exists k : Z, x = IZR k * PI
sin x = 0
  elim H; intros.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
sin x = 0
  apply (Zcase_sign x0).x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
x0 = 0%Z -> sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 > 0)%Z -> sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  intro.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:x0 = 0%Z
sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 > 0)%Z -> sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite H1 in H0.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = 0 * PI
H1:x0 = 0%Z
sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 > 0)%Z -> sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  simpl in H0.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = 0 * PI
H1:x0 = 0%Z
sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 > 0)%Z -> sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite H0; rewrite Rmult_0_l; apply sin_0.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 > 0)%Z -> sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  intro.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  cut (0 <= x0)%Z.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z -> sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  intro.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  elim (IZN x0 H2); intros.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
x1:nat
H3:x0 = Z.of_nat x1
sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite H3 in H0.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = IZR (Z.of_nat x1) * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite <- INR_IZR_INZ in H0.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
sin x = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite H0.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  elim (even_odd_cor x1); intros.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  elim H4; intro.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = (2 * x2)%nat
sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite H5.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = (2 * x2)%nat
sin (INR (2 * x2) * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite mult_INR.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = (2 * x2)%nat
sin (INR 2 * INR x2 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  simpl in |- *.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = (2 * x2)%nat
sin ((1 + 1) * INR x2 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite <- (Rplus_0_l ((1 + 1) * INR x2 * PI)).x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = (2 * x2)%nat
sin (0 + (1 + 1) * INR x2 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite sin_period.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = (2 * x2)%nat
sin 0 = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  apply sin_0.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite H5.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin (INR (S (2 * x2)) * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite S_INR; rewrite mult_INR.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin ((INR 2 * INR x2 + 1) * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  simpl in |- *.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin (((1 + 1) * INR x2 + 1) * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite Rmult_plus_distr_r.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin ((1 + 1) * INR x2 * PI + 1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite Rmult_1_l; rewrite sin_plus.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin ((1 + 1) * INR x2 * PI) * cos PI +
cos ((1 + 1) * INR x2 * PI) * sin PI = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite sin_PI.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin ((1 + 1) * INR x2 * PI) * cos PI +
cos ((1 + 1) * INR x2 * PI) * 0 = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite Rmult_0_r.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin ((1 + 1) * INR x2 * PI) * cos PI + 0 = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite <- (Rplus_0_l ((1 + 1) * INR x2 * PI)).x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin (0 + (1 + 1) * INR x2 * PI) * cos PI + 0 = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite sin_period.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
x1:nat
H0:x = INR x1 * PI
H1:(x0 > 0)%Z
H2:(0 <= x0)%Z
H3:x0 = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
sin 0 * cos PI + 0 = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  rewrite sin_0; ring.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 <= x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  apply le_IZR.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
0 <= IZR x0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  left; apply IZR_lt.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
(0 < x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  assert (H2 := Z.gt_lt_iff).x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
H2:forall n m : Z, (n > m)%Z <-> (m < n)%Z
(0 < x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  elim (H2 x0 0%Z); intros.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 > 0)%Z
H2:forall n m : Z, (n > m)%Z <-> (m < n)%Z
H3:(x0 > 0)%Z -> (0 < x0)%Z
H4:(0 < x0)%Z -> (x0 > 0)%Z
(0 < x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  apply H3; assumption.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
(x0 < 0)%Z -> sin x = 0
  intro.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
sin x = 0
  rewrite H0.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
sin (IZR x0 * PI) = 0
  replace (sin (IZR x0 * PI)) with (- sin (- IZR x0 * PI)).x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  cut (0 <= - x0)%Z.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z -> - sin (- IZR x0 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  intro.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
- sin (- IZR x0 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite <- Ropp_Ropp_IZR.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
- sin (IZR (- x0) * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  elim (IZN (- x0) H2); intros.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
- sin (IZR (- x0) * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite H3.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
- sin (IZR (Z.of_nat x1) * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite <- INR_IZR_INZ.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
- sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  elim (even_odd_cor x1); intros.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
- sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  elim H4; intro.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = (2 * x2)%nat
- sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite H5.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = (2 * x2)%nat
- sin (INR (2 * x2) * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite mult_INR.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = (2 * x2)%nat
- sin (INR 2 * INR x2 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  simpl in |- *.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = (2 * x2)%nat
- sin ((1 + 1) * INR x2 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite <- (Rplus_0_l ((1 + 1) * INR x2 * PI)).x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = (2 * x2)%nat
- sin (0 + (1 + 1) * INR x2 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite sin_period.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = (2 * x2)%nat
- sin 0 = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite sin_0; ring.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- sin (INR x1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite H5.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- sin (INR (S (2 * x2)) * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite S_INR; rewrite mult_INR.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- sin ((INR 2 * INR x2 + 1) * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  simpl in |- *.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- sin (((1 + 1) * INR x2 + 1) * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite Rmult_plus_distr_r.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- sin ((1 + 1) * INR x2 * PI + 1 * PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite Rmult_1_l; rewrite sin_plus.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
-
(sin ((1 + 1) * INR x2 * PI) * cos PI +
 cos ((1 + 1) * INR x2 * PI) * sin PI) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite sin_PI.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
-
(sin ((1 + 1) * INR x2 * PI) * cos PI +
 cos ((1 + 1) * INR x2 * PI) * 0) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite Rmult_0_r.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- (sin ((1 + 1) * INR x2 * PI) * cos PI + 0) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite <- (Rplus_0_l ((1 + 1) * INR x2 * PI)).x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- (sin (0 + (1 + 1) * INR x2 * PI) * cos PI + 0) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite sin_period.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
H2:(0 <= - x0)%Z
x1:nat
H3:(- x0)%Z = Z.of_nat x1
x2:nat
H4:x1 = (2 * x2)%nat \/ x1 = S (2 * x2)
H5:x1 = S (2 * x2)
- (sin 0 * cos PI + 0) = 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite sin_0; ring.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
(0 <= - x0)%Z
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  apply le_IZR.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
0 <= IZR (- x0)
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  apply Rplus_le_reg_l with (IZR x0).x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
IZR x0 + 0 <= IZR x0 + IZR (- x0)
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite Rplus_0_r.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
IZR x0 <= IZR x0 + IZR (- x0)
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite Ropp_Ropp_IZR.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
IZR x0 <= IZR x0 + - IZR x0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite Rplus_opp_r.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
IZR x0 <= 0
x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  now apply Rlt_le, IZR_lt.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
- sin (- IZR x0 * PI) = sin (IZR x0 * PI)
  rewrite <- sin_neg.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
sin (- (- IZR x0 * PI)) = sin (IZR x0 * PI)
  rewrite Ropp_mult_distr_l_reverse.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
sin (- - (IZR x0 * PI)) = sin (IZR x0 * PI)
  rewrite Ropp_involutive.x:R
H:exists k : Z, x = IZR k * PI
x0:Z
H0:x = IZR x0 * PI
H1:(x0 < 0)%Z
sin (IZR x0 * PI) = sin (IZR x0 * PI)
  reflexivity.
Qed.

Lemma sin_eq_0_0 (x:R) : sin x = 0 ->  exists k : Z, x = IZR k * PI.x:R
sin x = 0 -> exists k : Z, x = IZR k * PI
Proof.x:R
sin x = 0 -> exists k : Z, x = IZR k * PI
  intros Hx.x:R
Hx:sin x = 0
exists k : Z, x = IZR k * PI
  destruct (euclidian_division x PI PI_neq0) as (q & r & EQ & Hr & Hr').x:R
Hx:sin x = 0
q:Z
r:R
EQ:x = IZR q * PI + r
Hr:0 <= r
Hr':r < Rabs PI
exists k : Z, x = IZR k * PI
  exists q.x:R
Hx:sin x = 0
q:Z
r:R
EQ:x = IZR q * PI + r
Hr:0 <= r
Hr':r < Rabs PI
x = IZR q * PI
  rewrite <- (Rplus_0_r (_*_)).x:R
Hx:sin x = 0
q:Z
r:R
EQ:x = IZR q * PI + r
Hr:0 <= r
Hr':r < Rabs PI
x = IZR q * PI + 0 subst.q:Z
r:R
Hx:sin (IZR q * PI + r) = 0
Hr:0 <= r
Hr':r < Rabs PI
IZR q * PI + r = IZR q * PI + 0 apply Rplus_eq_compat_l.q:Z
r:R
Hx:sin (IZR q * PI + r) = 0
Hr:0 <= r
Hr':r < Rabs PI
r = 0
  rewrite sin_plus in Hx.q:Z
r:R
Hx:sin (IZR q * PI) * cos r +
cos (IZR q * PI) * sin r = 0
Hr:0 <= r
Hr':r < Rabs PI
r = 0
  assert (H : sin (IZR q * PI) = 0) by (apply sin_eq_0_1; now exists q).q:Z
r:R
Hx:sin (IZR q * PI) * cos r +
cos (IZR q * PI) * sin r = 0
Hr:0 <= r
Hr':r < Rabs PI
H:sin (IZR q * PI) = 0
r = 0
  rewrite H, Rmult_0_l, Rplus_0_l in Hx.q:Z
r:R
Hx:cos (IZR q * PI) * sin r = 0
Hr:0 <= r
Hr':r < Rabs PI
H:sin (IZR q * PI) = 0
r = 0
  destruct (Rmult_integral _ _ Hx) as [H'|H'].q:Z
r:R
Hx:cos (IZR q * PI) * sin r = 0
Hr:0 <= r
Hr':r < Rabs PI
H:sin (IZR q * PI) = 0
H':cos (IZR q * PI) = 0
r = 0
q:Z
r:R
Hx:cos (IZR q * PI) * sin r = 0
Hr:0 <= r
Hr':r < Rabs PI
H:sin (IZR q * PI) = 0
H':sin r = 0
r = 0
  -q:Z
r:R
Hx:cos (IZR q * PI) * sin r = 0
Hr:0 <= r
Hr':r < Rabs PI
H:sin (IZR q * PI) = 0
H':cos (IZR q * PI) = 0
r = 0 exfalso.q:Z
r:R
Hx:cos (IZR q * PI) * sin r = 0
Hr:0 <= r
Hr':r < Rabs PI
H:sin (IZR q * PI) = 0
H':cos (IZR q * PI) = 0
False
    generalize (sin2_cos2 (IZR q * PI)).q:Z
r:R
Hx:cos (IZR q * PI) * sin r = 0
Hr:0 <= r
Hr':r < Rabs PI
H:sin (IZR q * PI) = 0
H':cos (IZR q * PI) = 0
(sin (IZR q * PI))² + (cos (IZR q * PI))² = 1 -> False
    rewrite H, H', Rsqr_0, Rplus_0_l.q:Z
r:R
Hx:cos (IZR q * PI) * sin r = 0
Hr:0 <= r
Hr':r < Rabs PI
H:sin (IZR q * PI) = 0
H':cos (IZR q * PI) = 0
0 = 1 -> False
    intros; now apply R1_neq_R0.
  -q:Z
r:R
Hx:cos (IZR q * PI) * sin r = 0
Hr:0 <= r
Hr':r < Rabs PI
H:sin (IZR q * PI) = 0
H':sin r = 0
r = 0 rewrite Rabs_right in Hr'; [|left; apply PI_RGT_0].q:Z
r:R
Hx:cos (IZR q * PI) * sin r = 0
Hr:0 <= r
Hr':r < PI
H:sin (IZR q * PI) = 0
H':sin r = 0
r = 0
    destruct Hr as [Hr | ->]; trivial.q:Z
r:R
Hx:cos (IZR q * PI) * sin r = 0
Hr:0 < r
Hr':r < PI
H:sin (IZR q * PI) = 0
H':sin r = 0
r = 0
    exfalso.q:Z
r:R
Hx:cos (IZR q * PI) * sin r = 0
Hr:0 < r
Hr':r < PI
H:sin (IZR q * PI) = 0
H':sin r = 0
False
    generalize (sin_gt_0 r Hr Hr').q:Z
r:R
Hx:cos (IZR q * PI) * sin r = 0
Hr:0 < r
Hr':r < PI
H:sin (IZR q * PI) = 0
H':sin r = 0
0 < sin r -> False rewrite H'.q:Z
r:R
Hx:cos (IZR q * PI) * sin r = 0
Hr:0 < r
Hr':r < PI
H:sin (IZR q * PI) = 0
H':sin r = 0
0 < 0 -> False apply Rlt_irrefl.
Qed.

Lemma cos_eq_0_0 (x:R) :
  cos x = 0 ->  exists k : Z, x = IZR k * PI + PI / 2.x:R
cos x = 0 -> exists k : Z, x = IZR k * PI + PI / 2
Proof.x:R
cos x = 0 -> exists k : Z, x = IZR k * PI + PI / 2
  rewrite cos_sin.x:R
sin (PI / 2 + x) = 0 ->
exists k : Z, x = IZR k * PI + PI / 2 intros Hx.x:R
Hx:sin (PI / 2 + x) = 0
exists k : Z, x = IZR k * PI + PI / 2
  destruct (sin_eq_0_0 (PI/2 + x) Hx) as (k,Hk).x:R
Hx:sin (PI / 2 + x) = 0
k:Z
Hk:PI / 2 + x = IZR k * PI
exists k0 : Z, x = IZR k0 * PI + PI / 2 clear Hx.x:R
k:Z
Hk:PI / 2 + x = IZR k * PI
exists k0 : Z, x = IZR k0 * PI + PI / 2
  exists (k-1)%Z.x:R
k:Z
Hk:PI / 2 + x = IZR k * PI
x = IZR (k - 1) * PI + PI / 2 rewrite <- Z_R_minus; simpl.x:R
k:Z
Hk:PI / 2 + x = IZR k * PI
x = (IZR k - 1) * PI + PI / 2
  symmetry in Hk.x:R
k:Z
Hk:IZR k * PI = PI / 2 + x
x = (IZR k - 1) * PI + PI / 2 field_simplify [Hk].x:R
k:Z
Hk:IZR k * PI = PI / 2 + x
x = (- PI + 2 * (PI / 2) + 2 * x) / 2 field.
Qed.

Lemma cos_eq_0_1 (x:R) :
  (exists k : Z, x = IZR k * PI + PI / 2) -> cos x = 0.x:R
(exists k : Z, x = IZR k * PI + PI / 2) -> cos x = 0
Proof.x:R
(exists k : Z, x = IZR k * PI + PI / 2) -> cos x = 0
  rewrite cos_sin.x:R
(exists k : Z, x = IZR k * PI + PI / 2) ->
sin (PI / 2 + x) = 0 intros (k,->).k:Z
sin (PI / 2 + (IZR k * PI + PI / 2)) = 0
  replace (_ + _) with (IZR k * PI + PI) by field.k:Z
sin (IZR k * PI + PI) = 0
  rewrite neg_sin, <- Ropp_0.k:Z
- sin (IZR k * PI) = - 0 apply Ropp_eq_compat.k:Z
sin (IZR k * PI) = 0
  apply sin_eq_0_1.k:Z
exists k0 : Z, IZR k * PI = IZR k0 * PI now exists k.
Qed.

Lemma sin_eq_O_2PI_0 (x:R) :
  0 <= x -> x <= 2 * PI -> sin x = 0 ->
  x = 0 \/ x = PI \/ x = 2 * PI.x:R
0 <= x ->
x <= 2 * PI ->
sin x = 0 -> x = 0 \/ x = PI \/ x = 2 * PI
Proof.x:R
0 <= x ->
x <= 2 * PI ->
sin x = 0 -> x = 0 \/ x = PI \/ x = 2 * PI
  intros Lo Hi Hx.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin x = 0
x = 0 \/ x = PI \/ x = 2 * PI destruct (sin_eq_0_0 x Hx) as (k,Hk).x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin x = 0
k:Z
Hk:x = IZR k * PI
x = 0 \/ x = PI \/ x = 2 * PI clear Hx.x:R
Lo:0 <= x
Hi:x <= 2 * PI
k:Z
Hk:x = IZR k * PI
x = 0 \/ x = PI \/ x = 2 * PI
  destruct (Rtotal_order PI x) as [Hx|[Hx|Hx]].x:R
Lo:0 <= x
Hi:x <= 2 * PI
k:Z
Hk:x = IZR k * PI
Hx:PI < x
x = 0 \/ x = PI \/ x = 2 * PI
x:R
Lo:0 <= x
Hi:x <= 2 * PI
k:Z
Hk:x = IZR k * PI
Hx:PI = x
x = 0 \/ x = PI \/ x = 2 * PI
x:R
Lo:0 <= x
Hi:x <= 2 * PI
k:Z
Hk:x = IZR k * PI
Hx:PI > x
x = 0 \/ x = PI \/ x = 2 * PI
  -x:R
Lo:0 <= x
Hi:x <= 2 * PI
k:Z
Hk:x = IZR k * PI
Hx:PI < x
x = 0 \/ x = PI \/ x = 2 * PI right; right.x:R
Lo:0 <= x
Hi:x <= 2 * PI
k:Z
Hk:x = IZR k * PI
Hx:PI < x
x = 2 * PI
    clear Lo.x:R
Hi:x <= 2 * PI
k:Z
Hk:x = IZR k * PI
Hx:PI < x
x = 2 * PI subst.k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
IZR k * PI = 2 * PI
    f_equal.k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
IZR k = 2 change 2 with (IZR (- (-2))).k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
IZR k = IZR (- -2) f_equal.k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
k = (- -2)%Z
    apply Z.add_move_0_l.k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
(-2 + k)%Z = 0%Z
    apply one_IZR_lt1.k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
-1 < IZR (-2 + k) < 1
    rewrite plus_IZR; simpl.k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
-1 < -2 + IZR k < 1
    split.k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
-1 < -2 + IZR k
k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
-2 + IZR k < 1
    +k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
-1 < -2 + IZR k replace (-1) with (-2 + 1) by ring.k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
-2 + 1 < -2 + IZR k
      apply Rplus_lt_compat_l.k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
1 < IZR k
      apply Rmult_lt_reg_r with PI; [apply PI_RGT_0|].k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
1 * PI < IZR k * PI
      now rewrite Rmult_1_l.
    +k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
-2 + IZR k < 1 apply Rle_lt_trans with 0; [|apply Rlt_0_1].k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
-2 + IZR k <= 0
      replace 0 with (-2 + 2) by ring.k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
-2 + IZR k <= -2 + 2
      apply Rplus_le_compat_l.k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
IZR k <= 2
      apply Rmult_le_reg_r with PI; [apply PI_RGT_0|].k:Z
Hi:IZR k * PI <= 2 * PI
Hx:PI < IZR k * PI
IZR k * PI <= 2 * PI
      trivial.
  -x:R
Lo:0 <= x
Hi:x <= 2 * PI
k:Z
Hk:x = IZR k * PI
Hx:PI = x
x = 0 \/ x = PI \/ x = 2 * PI right; left; auto.
  -x:R
Lo:0 <= x
Hi:x <= 2 * PI
k:Z
Hk:x = IZR k * PI
Hx:PI > x
x = 0 \/ x = PI \/ x = 2 * PI left.x:R
Lo:0 <= x
Hi:x <= 2 * PI
k:Z
Hk:x = IZR k * PI
Hx:PI > x
x = 0
    clear Hi.x:R
Lo:0 <= x
k:Z
Hk:x = IZR k * PI
Hx:PI > x
x = 0 subst.k:Z
Lo:0 <= IZR k * PI
Hx:PI > IZR k * PI
IZR k * PI = 0
    replace 0 with (IZR 0 * PI) by apply Rmult_0_l.k:Z
Lo:0 <= IZR k * PI
Hx:PI > IZR k * PI
IZR k * PI = 0 * PI f_equal.k:Z
Lo:0 <= IZR k * PI
Hx:PI > IZR k * PI
IZR k = 0 f_equal.k:Z
Lo:0 <= IZR k * PI
Hx:PI > IZR k * PI
k = 0%Z
    apply one_IZR_lt1.k:Z
Lo:0 <= IZR k * PI
Hx:PI > IZR k * PI
-1 < IZR k < 1
    split.k:Z
Lo:0 <= IZR k * PI
Hx:PI > IZR k * PI
-1 < IZR k
k:Z
Lo:0 <= IZR k * PI
Hx:PI > IZR k * PI
IZR k < 1
    +k:Z
Lo:0 <= IZR k * PI
Hx:PI > IZR k * PI
-1 < IZR k apply Rlt_le_trans with 0;
       [rewrite <- Ropp_0; apply Ropp_gt_lt_contravar, Rlt_0_1 | ].k:Z
Lo:0 <= IZR k * PI
Hx:PI > IZR k * PI
0 <= IZR k
      apply Rmult_le_reg_r with PI; [apply PI_RGT_0|].k:Z
Lo:0 <= IZR k * PI
Hx:PI > IZR k * PI
0 * PI <= IZR k * PI
      now rewrite Rmult_0_l.
    +k:Z
Lo:0 <= IZR k * PI
Hx:PI > IZR k * PI
IZR k < 1 apply Rmult_lt_reg_r with PI; [apply PI_RGT_0|].k:Z
Lo:0 <= IZR k * PI
Hx:PI > IZR k * PI
IZR k * PI < 1 * PI
      now rewrite Rmult_1_l.
Qed.

Lemma sin_eq_O_2PI_1 (x:R) :
  0 <= x -> x <= 2 * PI ->
  x = 0 \/ x = PI \/ x = 2 * PI -> sin x = 0.x:R
0 <= x ->
x <= 2 * PI ->
x = 0 \/ x = PI \/ x = 2 * PI -> sin x = 0
Proof.x:R
0 <= x ->
x <= 2 * PI ->
x = 0 \/ x = PI \/ x = 2 * PI -> sin x = 0
  intros _ _ [ -> |[ -> | -> ]].sin 0 = 0
sin PI = 0
sin (2 * PI) = 0
  -sin 0 = 0 now rewrite sin_0.
  -sin PI = 0 now rewrite sin_PI.
  -sin (2 * PI) = 0 now rewrite sin_2PI.
Qed.

Lemma cos_eq_0_2PI_0 (x:R) :
  0 <= x -> x <= 2 * PI -> cos x = 0 ->
  x = PI / 2 \/ x = 3 * (PI / 2).x:R
0 <= x ->
x <= 2 * PI ->
cos x = 0 -> x = PI / 2 \/ x = 3 * (PI / 2)
Proof.x:R
0 <= x ->
x <= 2 * PI ->
cos x = 0 -> x = PI / 2 \/ x = 3 * (PI / 2)
  intros Lo Hi Hx.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:cos x = 0
x = PI / 2 \/ x = 3 * (PI / 2)
  destruct (Rtotal_order x (3 * (PI / 2))) as [LT|[EQ|GT]].x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:cos x = 0
LT:x < 3 * (PI / 2)
x = PI / 2 \/ x = 3 * (PI / 2)
x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:cos x = 0
EQ:x = 3 * (PI / 2)
x = PI / 2 \/ x = 3 * (PI / 2)
x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:cos x = 0
GT:x > 3 * (PI / 2)
x = PI / 2 \/ x = 3 * (PI / 2)
  -x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:cos x = 0
LT:x < 3 * (PI / 2)
x = PI / 2 \/ x = 3 * (PI / 2) rewrite cos_sin in Hx.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
x = PI / 2 \/ x = 3 * (PI / 2)
    assert (Lo' : 0 <= PI / 2 + x).x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
0 <= PI / 2 + x
x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
x = PI / 2 \/ x = 3 * (PI / 2)
    {x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
0 <= PI / 2 + x apply Rplus_le_le_0_compat.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
0 <= PI / 2
x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
0 <= x apply Rlt_le, PI2_RGT_0.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
0 <= x trivial. }x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
x = PI / 2 \/ x = 3 * (PI / 2)
    assert (Hi' : PI / 2 + x <= 2 * PI).x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
PI / 2 + x <= 2 * PI
x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
x = PI / 2 \/ x = 3 * (PI / 2)
    {x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
PI / 2 + x <= 2 * PI apply Rlt_le.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
PI / 2 + x < 2 * PI
      replace (2 * PI) with (PI / 2 + 3 * (PI / 2)) by field.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
PI / 2 + x < PI / 2 + 3 * (PI / 2)
      now apply Rplus_lt_compat_l. }x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
x = PI / 2 \/ x = 3 * (PI / 2)
    destruct (sin_eq_O_2PI_0 (PI / 2 + x) Lo' Hi' Hx) as [H|[H|H]].x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:PI / 2 + x = 0
x = PI / 2 \/ x = 3 * (PI / 2)
x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:PI / 2 + x = PI
x = PI / 2 \/ x = 3 * (PI / 2)
x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:PI / 2 + x = 2 * PI
x = PI / 2 \/ x = 3 * (PI / 2)
    +x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:PI / 2 + x = 0
x = PI / 2 \/ x = 3 * (PI / 2) exfalso.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:PI / 2 + x = 0
False
      apply (Rplus_le_compat_l (PI/2)) in Lo.x:R
Lo:PI / 2 + 0 <= PI / 2 + x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:PI / 2 + x = 0
False
      rewrite Rplus_0_r, H in Lo.x:R
Lo:PI / 2 <= 0
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:PI / 2 + x = 0
False
      apply (Rlt_irrefl 0 (Rlt_le_trans 0 (PI / 2) 0 PI2_RGT_0 Lo)).
    +x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:PI / 2 + x = PI
x = PI / 2 \/ x = 3 * (PI / 2) left.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:PI / 2 + x = PI
x = PI / 2
      apply (Rplus_eq_compat_l (-(PI/2))) in H.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:- (PI / 2) + (PI / 2 + x) = - (PI / 2) + PI
x = PI / 2
      ring_simplify in H.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:x = - (PI / 2) + PI
x = PI / 2 rewrite H.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:x = - (PI / 2) + PI
- (PI / 2) + PI = PI / 2 field.
    +x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:PI / 2 + x = 2 * PI
x = PI / 2 \/ x = 3 * (PI / 2) right.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:PI / 2 + x = 2 * PI
x = 3 * (PI / 2)
      apply (Rplus_eq_compat_l (-(PI/2))) in H.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:- (PI / 2) + (PI / 2 + x) = - (PI / 2) + 2 * PI
x = 3 * (PI / 2)
      ring_simplify in H.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:x = - (PI / 2) + 2 * PI
x = 3 * (PI / 2) rewrite H.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:sin (PI / 2 + x) = 0
LT:x < 3 * (PI / 2)
Lo':0 <= PI / 2 + x
Hi':PI / 2 + x <= 2 * PI
H:x = - (PI / 2) + 2 * PI
- (PI / 2) + 2 * PI = 3 * (PI / 2) field.
  -x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:cos x = 0
EQ:x = 3 * (PI / 2)
x = PI / 2 \/ x = 3 * (PI / 2) now right.
  -x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:cos x = 0
GT:x > 3 * (PI / 2)
x = PI / 2 \/ x = 3 * (PI / 2) exfalso.x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:cos x = 0
GT:x > 3 * (PI / 2)
False
    destruct (cos_eq_0_0 x Hx) as (k,Hk).x:R
Lo:0 <= x
Hi:x <= 2 * PI
Hx:cos x = 0
GT:x > 3 * (PI / 2)
k:Z
Hk:x = IZR k * PI + PI / 2
False clear Hx Lo.x:R
Hi:x <= 2 * PI
GT:x > 3 * (PI / 2)
k:Z
Hk:x = IZR k * PI + PI / 2
False
    subst.k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
False
    assert (LT : (k < 2)%Z).k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
(k < 2)%Z
k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
LT:(k < 2)%Z
False
    {k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
(k < 2)%Z apply lt_IZR.k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
IZR k < 2 simpl.k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
IZR k < 2
      apply (Rmult_lt_reg_r PI); [apply PI_RGT_0|].k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
IZR k * PI < 2 * PI
      apply Rlt_le_trans with (IZR k * PI + PI/2); trivial.k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
IZR k * PI < IZR k * PI + PI / 2
      rewrite <- (Rplus_0_r (IZR k * PI)) at 1.k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
IZR k * PI + 0 < IZR k * PI + PI / 2
      apply Rplus_lt_compat_l.k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
0 < PI / 2 apply PI2_RGT_0. }k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
LT:(k < 2)%Z
False
    assert (GT' : (1 < k)%Z).k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
LT:(k < 2)%Z
(1 < k)%Z
k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
LT:(k < 2)%Z
GT':(1 < k)%Z
False
    {k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
LT:(k < 2)%Z
(1 < k)%Z apply lt_IZR.k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
LT:(k < 2)%Z
1 < IZR k simpl.k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
LT:(k < 2)%Z
1 < IZR k
      apply (Rmult_lt_reg_r PI); [apply PI_RGT_0|rewrite Rmult_1_l].k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
LT:(k < 2)%Z
PI < IZR k * PI
      replace (3*(PI/2)) with (PI/2 + PI) in GT by field.k:Z
GT:IZR k * PI + PI / 2 > PI / 2 + PI
Hi:IZR k * PI + PI / 2 <= 2 * PI
LT:(k < 2)%Z
PI < IZR k * PI
      rewrite Rplus_comm in GT.k:Z
GT:PI / 2 + IZR k * PI > PI / 2 + PI
Hi:IZR k * PI + PI / 2 <= 2 * PI
LT:(k < 2)%Z
PI < IZR k * PI
      now apply Rplus_lt_reg_l in GT. }k:Z
GT:IZR k * PI + PI / 2 > 3 * (PI / 2)
Hi:IZR k * PI + PI / 2 <= 2 * PI
LT:(k < 2)%Z
GT':(1 < k)%Z
False
    omega.
Qed.

Lemma cos_eq_0_2PI_1 (x:R) :
  0 <= x -> x <= 2 * PI ->
  x = PI / 2 \/ x = 3 * (PI / 2) -> cos x = 0.x:R
0 <= x ->
x <= 2 * PI ->
x = PI / 2 \/ x = 3 * (PI / 2) -> cos x = 0
Proof.x:R
0 <= x ->
x <= 2 * PI ->
x = PI / 2 \/ x = 3 * (PI / 2) -> cos x = 0
 intros Lo Hi [ -> | -> ].Hi:PI / 2 <= 2 * PI
Lo:0 <= PI / 2
cos (PI / 2) = 0
Hi:3 * (PI / 2) <= 2 * PI
Lo:0 <= 3 * (PI / 2)
cos (3 * (PI / 2)) = 0
 -Hi:PI / 2 <= 2 * PI
Lo:0 <= PI / 2
cos (PI / 2) = 0 now rewrite cos_PI2.
 -Hi:3 * (PI / 2) <= 2 * PI
Lo:0 <= 3 * (PI / 2)
cos (3 * (PI / 2)) = 0 now rewrite cos_3PI2.
Qed.