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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 Lra.
Require Import Rbase.
Require Import PSeries_reg.
Require Import Rtrigo1.
Require Import Ranalysis_reg.
Require Import Rfunctions.
Require Import AltSeries.
Require Import Rseries.
Require Import SeqProp.
Require Import Ranalysis5.
Require Import SeqSeries.
Require Import PartSum.
Require Import Omega.

Local Open Scope R_scope.

Tools

Lemma Ropp_div : forall x y, -x/y = -(x/y).forall x y : R, - x / y = - (x / y)
Proof.forall x y : R, - x / y = - (x / y)
intros x y; unfold Rdiv; rewrite <-Ropp_mult_distr_l_reverse; reflexivity.
Qed.

Definition pos_half_prf : 0 < /2.0 < / 2
Proof.0 < / 2 lra. Qed.

Definition pos_half := mkposreal (/2) pos_half_prf.

Lemma Boule_half_to_interval :
  forall x , Boule (/2) pos_half x -> 0 <= x <= 1.forall x : R, Boule (/ 2) pos_half x -> 0 <= x <= 1
Proof.forall x : R, Boule (/ 2) pos_half x -> 0 <= x <= 1
unfold Boule, pos_half; simpl.forall x : R, Rabs (x - / 2) < / 2 -> 0 <= x <= 1
intros x b; apply Rabs_def2 in b; destruct b; split; lra.
Qed.

Lemma Boule_lt : forall c r x, Boule c r x -> Rabs x < Rabs c + r.forall (c : R) (r : posreal) (x : R),
Boule c r x -> Rabs x < Rabs c + r
Proof.forall (c : R) (r : posreal) (x : R),
Boule c r x -> Rabs x < Rabs c + r
unfold Boule; intros c r x h.c:R
r:posreal
x:R
h:Rabs (x - c) < r
Rabs x < Rabs c + r
apply Rabs_def2 in h; destruct h; apply Rabs_def1;
 (destruct (Rle_lt_dec 0 c);[rewrite Rabs_pos_eq; lra |
    rewrite <- Rabs_Ropp, Rabs_pos_eq; lra]).
Qed.

(* The following lemma does not belong here. *)
Lemma Un_cv_ext :
  forall un vn, (forall n, un n = vn n) ->
  forall l, Un_cv un l -> Un_cv vn l.forall un vn : nat -> R,
(forall n : nat, un n = vn n) ->
forall l : R, Un_cv un l -> Un_cv vn l
Proof.forall un vn : nat -> R,
(forall n : nat, un n = vn n) ->
forall l : R, Un_cv un l -> Un_cv vn l
intros un vn quv l P eps ep; destruct (P eps ep) as [N Pn]; exists N.un, vn:nat -> R
quv:forall n : nat, un n = vn n
l:R
P:Un_cv un l
eps:R
ep:eps > 0
N:nat
Pn:forall n : nat,
(n >= N)%nat -> R_dist (un n) l < eps
forall n : nat, (n >= N)%nat -> R_dist (vn n) l < eps
intro n; rewrite <- quv; apply Pn.
Qed.

(* The following two lemmas are general purposes about alternated series.
  They do not belong here. *)
Lemma Alt_first_term_bound :forall f l N n,
   Un_decreasing f -> Un_cv f 0 ->
   Un_cv (sum_f_R0 (tg_alt f)) l ->
   (N <= n)%nat ->
   R_dist (sum_f_R0 (tg_alt f) n) l <= f N.forall (f : nat -> R) (l : R) (N n : nat),
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(N <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f N
Proof.forall (f : nat -> R) (l : R) (N n : nat),
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(N <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f N
intros f l.f:nat -> R
l:R
forall N n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(N <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f N
assert (WLOG :
   forall n P, (forall k, (0 < k)%nat -> P k) ->
         ((forall k, (0 < k)%nat -> P k) -> P 0%nat) -> P n).f:nat -> R
l:R
forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) -> P 0%nat) ->
P n
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
forall N n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(N <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f N
clear.forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) -> P 0%nat) ->
P n
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
forall N n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(N <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f N
intros [ | n] P Hs Ho;[solve[apply Ho, Hs] | apply Hs; auto with arith].f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
forall N n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(N <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f N
intros N; pattern N; apply WLOG; clear N.f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
forall k : nat,
(0 < k)%nat ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f k
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
intros [ | N] Npos n decr to0 cv nN.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
Npos:(0 < 0)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(0 <= n)%nat
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  clear -Npos; elimtype False; omega.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 assert (decr' : Un_decreasing (fun i => f (S N + i)%nat)).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
Un_decreasing (fun i : nat => f (S N + i)%nat)
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  intros k; replace (S N+S k)%nat with (S (S N+k)) by ring.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k0 : nat, (0 < k0)%nat -> P k0) ->
((forall k0 : nat, (0 < k0)%nat -> P k0) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
k:nat
f (S (S N + k)) <= f (S N + k)%nat
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  apply (decr (S N + k)%nat).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 assert (to' : Un_cv (fun i => f (S N + i)%nat) 0).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
Un_cv (fun i : nat => f (S N + i)%nat) 0
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  intros eps ep; destruct (to0 eps ep) as [M PM].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat -> R_dist (f n0) 0 < eps
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> R_dist (f (S N + n0)%nat) 0 < eps
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  exists M; intros k kM; apply PM; omega.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 assert (cv' : Un_cv
         (sum_f_R0 (tg_alt (fun i => ((-1) ^ S N * f(S N + i)%nat))))
             (l - sum_f_R0 (tg_alt f) N)).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  intros eps ep; destruct (cv eps ep) as [M PM]; exists M.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
forall n0 : nat,
(n0 >= M)%nat ->
R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat))
     n0) (l - sum_f_R0 (tg_alt f) N) < eps
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  intros n' nM.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat))
     n') (l - sum_f_R0 (tg_alt f) N) < eps
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat 
  match goal with |- ?C => set (U := C) end.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
U:=R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)) n')
  (l - sum_f_R0 (tg_alt f) N) < eps:Prop
U
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  assert (nM' : (n' + S N >= M)%nat) by omega.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
U:=R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)) n')
  (l - sum_f_R0 (tg_alt f) N) < eps:Prop
nM':(n' + S N >= M)%nat
U
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   generalize (PM _ nM'); unfold R_dist.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
U:=R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)) n')
  (l - sum_f_R0 (tg_alt f) N) < eps:Prop
nM':(n' + S N >= M)%nat
Rabs (sum_f_R0 (tg_alt f) (n' + S N) - l) < eps -> U
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  rewrite (tech2 (tg_alt f) N (n' + S N)).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
U:=R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)) n')
  (l - sum_f_R0 (tg_alt f) N) < eps:Prop
nM':(n' + S N >= M)%nat
Rabs
  (sum_f_R0 (tg_alt f) N +
   sum_f_R0 (fun i : nat => tg_alt f (S N + i))
     (n' + S N - S N) - l) < eps -> U
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
U:=R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)) n')
  (l - sum_f_R0 (tg_alt f) N) < eps:Prop
nM':(n' + S N >= M)%nat
(N < n' + S N)%nat
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  assert (t : forall a b c, (a + b) - c = b - (c - a)) by (intros; ring).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
U:=R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)) n')
  (l - sum_f_R0 (tg_alt f) N) < eps:Prop
nM':(n' + S N >= M)%nat
t:forall a b c : R, a + b - c = b - (c - a)
Rabs
  (sum_f_R0 (tg_alt f) N +
   sum_f_R0 (fun i : nat => tg_alt f (S N + i))
     (n' + S N - S N) - l) < eps -> U
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
U:=R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)) n')
  (l - sum_f_R0 (tg_alt f) N) < eps:Prop
nM':(n' + S N >= M)%nat
(N < n' + S N)%nat
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  rewrite t; clear t; unfold U, R_dist; clear U.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
nM':(n' + S N >= M)%nat
Rabs
  (sum_f_R0 (fun i : nat => tg_alt f (S N + i))
     (n' + S N - S N) - (l - sum_f_R0 (tg_alt f) N)) <
eps ->
Rabs
  (sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat))
     n' - (l - sum_f_R0 (tg_alt f) N)) < eps
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
U:=R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)) n')
  (l - sum_f_R0 (tg_alt f) N) < eps:Prop
nM':(n' + S N >= M)%nat
(N < n' + S N)%nat
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  replace (n' + S N - S N)%nat with n' by omega.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
nM':(n' + S N >= M)%nat
Rabs
  (sum_f_R0 (fun i : nat => tg_alt f (S N + i)) n' -
   (l - sum_f_R0 (tg_alt f) N)) < eps ->
Rabs
  (sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat))
     n' - (l - sum_f_R0 (tg_alt f) N)) < eps
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
U:=R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)) n')
  (l - sum_f_R0 (tg_alt f) N) < eps:Prop
nM':(n' + S N >= M)%nat
(N < n' + S N)%nat
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  rewrite <- (sum_eq (tg_alt (fun i => (-1) ^ S N * f(S N + i)%nat))).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
nM':(n' + S N >= M)%nat
Rabs
  (sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat))
     n' - (l - sum_f_R0 (tg_alt f) N)) < eps ->
Rabs
  (sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat))
     n' - (l - sum_f_R0 (tg_alt f) N)) < eps
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
nM':(n' + S N >= M)%nat
forall i : nat,
(i <= n')%nat ->
tg_alt (fun i0 : nat => (-1) ^ S N * f (S N + i0)%nat)
  i = tg_alt f (S N + i)
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
U:=R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)) n')
  (l - sum_f_R0 (tg_alt f) N) < eps:Prop
nM':(n' + S N >= M)%nat
(N < n' + S N)%nat
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   tauto.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
nM':(n' + S N >= M)%nat
forall i : nat,
(i <= n')%nat ->
tg_alt (fun i0 : nat => (-1) ^ S N * f (S N + i0)%nat)
  i = tg_alt f (S N + i)
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
U:=R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)) n')
  (l - sum_f_R0 (tg_alt f) N) < eps:Prop
nM':(n' + S N >= M)%nat
(N < n' + S N)%nat
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   intros i _; unfold tg_alt.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing
  (fun i0 : nat => f (S N + i0)%nat)
to':Un_cv (fun i0 : nat => f (S N + i0)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
nM':(n' + S N >= M)%nat
i:nat
(-1) ^ i * ((-1) ^ S N * f (S N + i)%nat) =
(-1) ^ (S N + i) * f (S N + i)%nat
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
U:=R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)) n')
  (l - sum_f_R0 (tg_alt f) N) < eps:Prop
nM':(n' + S N >= M)%nat
(N < n' + S N)%nat
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   rewrite <- Rmult_assoc, <- pow_add, !(plus_comm i); reflexivity.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
eps:R
ep:eps > 0
M:nat
PM:forall n0 : nat,
(n0 >= M)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l < eps
n':nat
nM:(n' >= M)%nat
U:=R_dist
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)) n')
  (l - sum_f_R0 (tg_alt f) N) < eps:Prop
nM':(n' + S N >= M)%nat
(N < n' + S N)%nat
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  omega.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  assert (cv'' : Un_cv (sum_f_R0 (tg_alt (fun i => f (S N + i)%nat)))
                   ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  apply (Un_cv_ext (fun n => (-1) ^ S N * 
            sum_f_R0 (tg_alt (fun i : nat => (-1) ^ S N * f (S N + i)%nat)) n)).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
forall n0 : nat,
(-1) ^ S N *
sum_f_R0
  (tg_alt
     (fun i : nat => (-1) ^ S N * f (S N + i)%nat)) n0 =
sum_f_R0 (tg_alt (fun i : nat => f (S N + i)%nat)) n0
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
Un_cv
  (fun n0 : nat =>
   (-1) ^ S N *
   sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat))
     n0) ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   intros n0; rewrite scal_sum; apply sum_eq; intros i _.f:nat -> R
l:R
WLOG:forall (n1 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n1
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing
  (fun i0 : nat => f (S N + i0)%nat)
to':Un_cv (fun i0 : nat => f (S N + i0)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i0 : nat =>
         (-1) ^ S N * f (S N + i0)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
n0, i:nat
tg_alt (fun i0 : nat => (-1) ^ S N * f (S N + i0)%nat)
  i * (-1) ^ S N =
tg_alt (fun i0 : nat => f (S N + i0)%nat) i
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
Un_cv
  (fun n0 : nat =>
   (-1) ^ S N *
   sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat))
     n0) ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   unfold tg_alt; ring_simplify; replace (((-1) ^ S N) ^ 2) with 1.f:nat -> R
l:R
WLOG:forall (n1 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n1
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing
  (fun i0 : nat => f (S N + i0)%nat)
to':Un_cv (fun i0 : nat => f (S N + i0)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i0 : nat =>
         (-1) ^ S N * f (S N + i0)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
n0, i:nat
(-1) ^ i * 1 * f (S N + i)%nat =
(-1) ^ i * f (S N + i)%nat
f:nat -> R
l:R
WLOG:forall (n1 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n1
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing
  (fun i0 : nat => f (S N + i0)%nat)
to':Un_cv (fun i0 : nat => f (S N + i0)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i0 : nat =>
         (-1) ^ S N * f (S N + i0)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
n0, i:nat
1 = ((-1) ^ S N) ^ 2
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
Un_cv
  (fun n0 : nat =>
   (-1) ^ S N *
   sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat))
     n0) ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
    ring.f:nat -> R
l:R
WLOG:forall (n1 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n1
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing
  (fun i0 : nat => f (S N + i0)%nat)
to':Un_cv (fun i0 : nat => f (S N + i0)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i0 : nat =>
         (-1) ^ S N * f (S N + i0)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
n0, i:nat
1 = ((-1) ^ S N) ^ 2
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
Un_cv
  (fun n0 : nat =>
   (-1) ^ S N *
   sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat))
     n0) ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   rewrite <- pow_mult, mult_comm, pow_mult; replace ((-1) ^2) with 1 by ring.f:nat -> R
l:R
WLOG:forall (n1 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n1
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing
  (fun i0 : nat => f (S N + i0)%nat)
to':Un_cv (fun i0 : nat => f (S N + i0)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i0 : nat =>
         (-1) ^ S N * f (S N + i0)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
n0, i:nat
1 = 1 ^ S N
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
Un_cv
  (fun n0 : nat =>
   (-1) ^ S N *
   sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat))
     n0) ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   rewrite pow1; reflexivity.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
Un_cv
  (fun n0 : nat =>
   (-1) ^ S N *
   sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat))
     n0) ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  apply CV_mult.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
Un_cv (fun _ : nat => (-1) ^ S N) ((-1) ^ S N)
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   solve[intros eps ep; exists 0%nat; intros; rewrite R_dist_eq; auto].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat => (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  assumption.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 destruct (even_odd_cor N) as [p [Neven | Nodd]].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  rewrite Neven; destruct (alternated_series_ineq _ _ p decr to0 cv) as [B C].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  case (even_odd_cor n) as [p' [neven | nodd]].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
neven:n = (2 * p')%nat
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   rewrite neven.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
neven:n = (2 * p')%nat
R_dist (sum_f_R0 (tg_alt f) (2 * p')) l <=
f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   destruct (alternated_series_ineq _ _ p' decr to0 cv) as [D E].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
R_dist (sum_f_R0 (tg_alt f) (2 * p')) l <=
f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   unfold R_dist; rewrite Rabs_pos_eq;[ | lra].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
sum_f_R0 (tg_alt f) (2 * p') - l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   assert (dist : (p <= p')%nat) by omega.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(p <= p')%nat
sum_f_R0 (tg_alt f) (2 * p') - l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   assert (t := decreasing_prop _ _ _ (CV_ALT_step1 f decr) dist).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(p <= p')%nat
t:(fun N0 : nat => sum_f_R0 (tg_alt f) (2 * N0)) p' <=
(fun N0 : nat => sum_f_R0 (tg_alt f) (2 * N0)) p
sum_f_R0 (tg_alt f) (2 * p') - l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   apply Rle_trans with (sum_f_R0 (tg_alt f) (2 * p) - l).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(p <= p')%nat
t:(fun N0 : nat => sum_f_R0 (tg_alt f) (2 * N0)) p' <=
(fun N0 : nat => sum_f_R0 (tg_alt f) (2 * N0)) p
sum_f_R0 (tg_alt f) (2 * p') - l <=
sum_f_R0 (tg_alt f) (2 * p) - l
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(p <= p')%nat
t:(fun N0 : nat => sum_f_R0 (tg_alt f) (2 * N0)) p' <=
(fun N0 : nat => sum_f_R0 (tg_alt f) (2 * N0)) p
sum_f_R0 (tg_alt f) (2 * p) - l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
    unfold Rminus; apply Rplus_le_compat_r; exact t.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(p <= p')%nat
t:(fun N0 : nat => sum_f_R0 (tg_alt f) (2 * N0)) p' <=
(fun N0 : nat => sum_f_R0 (tg_alt f) (2 * N0)) p
sum_f_R0 (tg_alt f) (2 * p) - l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   match goal with _ : ?a <= l, _ : l <= ?b |- _ => 
    replace (f (S (2 * p))) with (b - a) by
     (rewrite tech5; unfold tg_alt; rewrite pow_1_odd; ring); lra
   end.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  rewrite nodd; destruct (alternated_series_ineq _ _ p' decr to0 cv) as [D E].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
R_dist (sum_f_R0 (tg_alt f) (S (2 * p'))) l <=
f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   unfold R_dist; rewrite <- Rabs_Ropp, Rabs_pos_eq, Ropp_minus_distr;
     [ | lra].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
l - sum_f_R0 (tg_alt f) (S (2 * p')) <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   assert (dist : (p <= p')%nat) by omega.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(p <= p')%nat
l - sum_f_R0 (tg_alt f) (S (2 * p')) <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   apply Rle_trans with (l - sum_f_R0 (tg_alt f) (S (2 * p))).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(p <= p')%nat
l - sum_f_R0 (tg_alt f) (S (2 * p')) <=
l - sum_f_R0 (tg_alt f) (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(p <= p')%nat
l - sum_f_R0 (tg_alt f) (S (2 * p)) <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
    unfold Rminus; apply Rplus_le_compat_l, Ropp_le_contravar.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(p <= p')%nat
sum_f_R0 (tg_alt f) (S (2 * p)) <=
sum_f_R0 (tg_alt f) (S (2 * p'))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(p <= p')%nat
l - sum_f_R0 (tg_alt f) (S (2 * p)) <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
    solve[apply Rge_le, (growing_prop _ _ _ (CV_ALT_step0 f decr) dist)].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(p <= p')%nat
l - sum_f_R0 (tg_alt f) (S (2 * p)) <= f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   unfold Rminus; rewrite tech5, Ropp_plus_distr, <- Rplus_assoc.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Neven:N = (2 * p)%nat
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * p)
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(p <= p')%nat
l + - sum_f_R0 (tg_alt f) (2 * p) +
- tg_alt f (S (2 * p)) <= 
f (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   unfold tg_alt at 2; rewrite pow_1_odd; lra.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S N)
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 rewrite Nodd; destruct (alternated_series_ineq _ _ p decr to0 cv) as [B _].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 destruct (alternated_series_ineq _ _ (S p) decr to0 cv) as [_ C].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 assert (keep : (2 * S p = S (S ( 2 * p)))%nat) by ring.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 case (even_odd_cor n) as [p' [neven | nodd]].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
neven:n = (2 * p')%nat
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   rewrite neven;
   destruct (alternated_series_ineq _ _ p' decr to0 cv) as [D E].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
R_dist (sum_f_R0 (tg_alt f) (2 * p')) l <=
f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   unfold R_dist; rewrite Rabs_pos_eq;[ | lra].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
sum_f_R0 (tg_alt f) (2 * p') - l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   assert (dist : (S p < S p')%nat) by omega.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(S p < S p')%nat
sum_f_R0 (tg_alt f) (2 * p') - l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   apply Rle_trans with (sum_f_R0 (tg_alt f) (2 * S p) - l).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(S p < S p')%nat
sum_f_R0 (tg_alt f) (2 * p') - l <=
sum_f_R0 (tg_alt f) (2 * S p) - l
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(S p < S p')%nat
sum_f_R0 (tg_alt f) (2 * S p) - l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   unfold Rminus; apply Rplus_le_compat_r,
     (decreasing_prop _ _ _ (CV_ALT_step1 f decr)).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(S p < S p')%nat
(S p <= p')%nat
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(S p < S p')%nat
sum_f_R0 (tg_alt f) (2 * S p) - l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
   omega.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(S p < S p')%nat
sum_f_R0 (tg_alt f) (2 * S p) - l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  rewrite keep, tech5; unfold tg_alt at 2; rewrite <- keep, pow_1_even.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
neven:n = (2 * p')%nat
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
dist:(S p < S p')%nat
sum_f_R0 (tg_alt f) (S (2 * p)) + 1 * f (2 * S p)%nat -
l <= f (2 * S p)%nat
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  lra.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
R_dist (sum_f_R0 (tg_alt f) n) l <= f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 rewrite nodd; destruct (alternated_series_ineq _ _ p' decr to0 cv) as [D E].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
R_dist (sum_f_R0 (tg_alt f) (S (2 * p'))) l <=
f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 unfold R_dist; rewrite <- Rabs_Ropp, Rabs_pos_eq;[ | lra].f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
- (sum_f_R0 (tg_alt f) (S (2 * p')) - l) <=
f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 rewrite Ropp_minus_distr.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
l - sum_f_R0 (tg_alt f) (S (2 * p')) <=
f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 apply Rle_trans with (l - sum_f_R0 (tg_alt f) (S (2 * p))).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
l - sum_f_R0 (tg_alt f) (S (2 * p')) <=
l - sum_f_R0 (tg_alt f) (S (2 * p))
f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
l - sum_f_R0 (tg_alt f) (S (2 * p)) <=
f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
  unfold Rminus; apply Rplus_le_compat_l, Ropp_le_contravar, Rge_le,
   (growing_prop _ _ _ (CV_ALT_step0 f decr)); omega.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
l - sum_f_R0 (tg_alt f) (S (2 * p)) <=
f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 generalize C; rewrite keep, tech5; unfold tg_alt.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
l <=
sum_f_R0 (fun i : nat => (-1) ^ i * f i) (S (2 * p)) +
(-1) ^ S (S (2 * p)) * f (S (S (2 * p))) ->
l -
sum_f_R0 (fun i : nat => (-1) ^ i * f i) (S (2 * p)) <=
f (S (S (2 * p)))
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 rewrite <- keep, pow_1_even.f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
l <=
sum_f_R0 (fun i : nat => (-1) ^ i * f i) (S (2 * p)) +
1 * f (2 * S p)%nat ->
l -
sum_f_R0 (fun i : nat => (-1) ^ i * f i) (S (2 * p)) <=
f (2 * S p)%nat
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 assert (t : forall a b c, a <= b + 1 * c -> a - b <= c) by (intros; lra).f:nat -> R
l:R
WLOG:forall (n0 : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n0
N:nat
Npos:(0 < S N)%nat
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
nN:(S N <= n)%nat
decr':Un_decreasing (fun i : nat => f (S N + i)%nat)
to':Un_cv (fun i : nat => f (S N + i)%nat) 0
cv':Un_cv
  (sum_f_R0
     (tg_alt
        (fun i : nat =>
         (-1) ^ S N * f (S N + i)%nat)))
  (l - sum_f_R0 (tg_alt f) N)
cv'':Un_cv
  (sum_f_R0
     (tg_alt (fun i : nat => f (S N + i)%nat)))
  ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))
p:nat
Nodd:N = S (2 * p)
B:sum_f_R0 (tg_alt f) (S (2 * p)) <= l
C:l <= sum_f_R0 (tg_alt f) (2 * S p)
keep:(2 * S p)%nat = S (S (2 * p))
p':nat
nodd:n = S (2 * p')
D:sum_f_R0 (tg_alt f) (S (2 * p')) <= l
E:l <= sum_f_R0 (tg_alt f) (2 * p')
t:forall a b c : R, a <= b + 1 * c -> a - b <= c
l <=
sum_f_R0 (fun i : nat => (-1) ^ i * f i) (S (2 * p)) +
1 * f (2 * S p)%nat ->
l -
sum_f_R0 (fun i : nat => (-1) ^ i * f i) (S (2 * p)) <=
f (2 * S p)%nat
f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
 solve[apply t].f:nat -> R
l:R
WLOG:forall (n : nat) (P : nat -> Type),
(forall k : nat, (0 < k)%nat -> P k) ->
((forall k : nat, (0 < k)%nat -> P k) ->
 P 0%nat) -> P n
(forall k : nat,
 (0 < k)%nat ->
 forall n : nat,
 Un_decreasing f ->
 Un_cv f 0 ->
 Un_cv (sum_f_R0 (tg_alt f)) l ->
 (k <= n)%nat ->
 R_dist (sum_f_R0 (tg_alt f) n) l <= f k) ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(0 <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f 0%nat
clear WLOG; intros Hyp [ | n] decr to0 cv _.f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f k
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
R_dist (sum_f_R0 (tg_alt f) 0) l <= f 0%nat
f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n0 : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n0)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l <= f k
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
R_dist (sum_f_R0 (tg_alt f) (S n)) l <= f 0%nat
 generalize (alternated_series_ineq f l 0 decr to0 cv).f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f k
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
sum_f_R0 (tg_alt f) (S (2 * 0)) <= l <=
sum_f_R0 (tg_alt f) (2 * 0) ->
R_dist (sum_f_R0 (tg_alt f) 0) l <= f 0%nat
f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n0 : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n0)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l <= f k
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
R_dist (sum_f_R0 (tg_alt f) (S n)) l <= f 0%nat
 unfold R_dist, tg_alt; simpl; rewrite !Rmult_1_l, !Rmult_1_r.f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f k
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
f 0%nat + -1 * f 1%nat <= l <= f 0%nat ->
Rabs (f 0%nat - l) <= f 0%nat
f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n0 : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n0)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l <= f k
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
R_dist (sum_f_R0 (tg_alt f) (S n)) l <= f 0%nat
 assert (f 1%nat <= f 0%nat) by apply decr.f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f k
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
H:f 1%nat <= f 0%nat
f 0%nat + -1 * f 1%nat <= l <= f 0%nat ->
Rabs (f 0%nat - l) <= f 0%nat
f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n0 : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n0)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l <= f k
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
R_dist (sum_f_R0 (tg_alt f) (S n)) l <= f 0%nat
 intros [A B]; rewrite Rabs_pos_eq; lra.f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n0 : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n0)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l <= f k
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
R_dist (sum_f_R0 (tg_alt f) (S n)) l <= f 0%nat
apply Rle_trans with (f 1%nat).f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n0 : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n0)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l <= f k
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
R_dist (sum_f_R0 (tg_alt f) (S n)) l <= f 1%nat
f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n0 : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n0)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l <= f k
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
f 1%nat <= f 0%nat
 apply (Hyp 1%nat (le_n 1) (S n) decr to0 cv).f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n0 : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n0)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l <= f k
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
(1 <= S n)%nat
f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n0 : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n0)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l <= f k
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
f 1%nat <= f 0%nat
 omega.f:nat -> R
l:R
Hyp:forall k : nat,
(0 < k)%nat ->
forall n0 : nat,
Un_decreasing f ->
Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(k <= n0)%nat ->
R_dist (sum_f_R0 (tg_alt f) n0) l <= f k
n:nat
decr:Un_decreasing f
to0:Un_cv f 0
cv:Un_cv (sum_f_R0 (tg_alt f)) l
f 1%nat <= f 0%nat
solve[apply decr].
Qed.

Lemma Alt_CVU : forall (f : nat -> R -> R) g h c r,
   (forall x, Boule c r x ->Un_decreasing (fun n => f n x)) -> 
   (forall x, Boule c r x -> Un_cv (fun n => f n x) 0) ->
   (forall x, Boule c r x -> 
       Un_cv (sum_f_R0 (tg_alt  (fun i => f i x))) (g x)) ->
   (forall x n, Boule c r x -> f n x <= h n) ->
   (Un_cv h 0) ->
   CVU (fun N x => sum_f_R0 (tg_alt (fun i => f i x)) N) g c r.forall (f : nat -> R -> R) (g : R -> R) (h : nat -> R)
  (c : R) (r : posreal),
(forall x : R,
 Boule c r x -> Un_decreasing (fun n : nat => f n x)) ->
(forall x : R,
 Boule c r x -> Un_cv (fun n : nat => f n x) 0) ->
(forall x : R,
 Boule c r x ->
 Un_cv (sum_f_R0 (tg_alt (fun i : nat => f i x)))
   (g x)) ->
(forall (x : R) (n : nat), Boule c r x -> f n x <= h n) ->
Un_cv h 0 ->
CVU
  (fun (N : nat) (x : R) =>
   sum_f_R0 (tg_alt (fun i : nat => f i x)) N) g c r
Proof.forall (f : nat -> R -> R) (g : R -> R) (h : nat -> R)
  (c : R) (r : posreal),
(forall x : R,
 Boule c r x -> Un_decreasing (fun n : nat => f n x)) ->
(forall x : R,
 Boule c r x -> Un_cv (fun n : nat => f n x) 0) ->
(forall x : R,
 Boule c r x ->
 Un_cv (sum_f_R0 (tg_alt (fun i : nat => f i x)))
   (g x)) ->
(forall (x : R) (n : nat), Boule c r x -> f n x <= h n) ->
Un_cv h 0 ->
CVU
  (fun (N : nat) (x : R) =>
   sum_f_R0 (tg_alt (fun i : nat => f i x)) N) g c r
intros f g h c r decr to0 to_g bound bound0 eps ep.f:nat -> R -> R
g:R -> R
h:nat -> R
c:R
r:posreal
decr:forall x : R,
Boule c r x ->
Un_decreasing (fun n : nat => f n x)
to0:forall x : R,
Boule c r x -> Un_cv (fun n : nat => f n x) 0
to_g:forall x : R,
Boule c r x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => f i x)))
  (g x)
bound:forall (x : R) (n : nat),
Boule c r x -> f n x <= h n
bound0:Un_cv h 0
eps:R
ep:0 < eps
exists N : nat,
  forall (n : nat) (y : R),
  (N <= n)%nat ->
  Boule c r y ->
  Rabs
    (g y - sum_f_R0 (tg_alt (fun i : nat => f i y)) n) <
  eps  
assert (ep' : 0 <eps/2) by lra.f:nat -> R -> R
g:R -> R
h:nat -> R
c:R
r:posreal
decr:forall x : R,
Boule c r x ->
Un_decreasing (fun n : nat => f n x)
to0:forall x : R,
Boule c r x -> Un_cv (fun n : nat => f n x) 0
to_g:forall x : R,
Boule c r x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => f i x)))
  (g x)
bound:forall (x : R) (n : nat),
Boule c r x -> f n x <= h n
bound0:Un_cv h 0
eps:R
ep:0 < eps
ep':0 < eps / 2
exists N : nat,
  forall (n : nat) (y : R),
  (N <= n)%nat ->
  Boule c r y ->
  Rabs
    (g y - sum_f_R0 (tg_alt (fun i : nat => f i y)) n) <
  eps
destruct (bound0 _ ep) as [N Pn]; exists N.f:nat -> R -> R
g:R -> R
h:nat -> R
c:R
r:posreal
decr:forall x : R,
Boule c r x ->
Un_decreasing (fun n : nat => f n x)
to0:forall x : R,
Boule c r x -> Un_cv (fun n : nat => f n x) 0
to_g:forall x : R,
Boule c r x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => f i x)))
  (g x)
bound:forall (x : R) (n : nat),
Boule c r x -> f n x <= h n
bound0:Un_cv h 0
eps:R
ep:0 < eps
ep':0 < eps / 2
N:nat
Pn:forall n : nat,
(n >= N)%nat -> R_dist (h n) 0 < eps
forall (n : nat) (y : R),
(N <= n)%nat ->
Boule c r y ->
Rabs
  (g y - sum_f_R0 (tg_alt (fun i : nat => f i y)) n) <
eps
intros n y nN dy.f:nat -> R -> R
g:R -> R
h:nat -> R
c:R
r:posreal
decr:forall x : R,
Boule c r x ->
Un_decreasing (fun n0 : nat => f n0 x)
to0:forall x : R,
Boule c r x -> Un_cv (fun n0 : nat => f n0 x) 0
to_g:forall x : R,
Boule c r x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => f i x)))
  (g x)
bound:forall (x : R) (n0 : nat),
Boule c r x -> f n0 x <= h n0
bound0:Un_cv h 0
eps:R
ep:0 < eps
ep':0 < eps / 2
N:nat
Pn:forall n0 : nat,
(n0 >= N)%nat -> R_dist (h n0) 0 < eps
n:nat
y:R
nN:(N <= n)%nat
dy:Boule c r y
Rabs
  (g y - sum_f_R0 (tg_alt (fun i : nat => f i y)) n) <
eps
rewrite <- Rabs_Ropp, Ropp_minus_distr; apply Rle_lt_trans with (f n y).f:nat -> R -> R
g:R -> R
h:nat -> R
c:R
r:posreal
decr:forall x : R,
Boule c r x ->
Un_decreasing (fun n0 : nat => f n0 x)
to0:forall x : R,
Boule c r x -> Un_cv (fun n0 : nat => f n0 x) 0
to_g:forall x : R,
Boule c r x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => f i x)))
  (g x)
bound:forall (x : R) (n0 : nat),
Boule c r x -> f n0 x <= h n0
bound0:Un_cv h 0
eps:R
ep:0 < eps
ep':0 < eps / 2
N:nat
Pn:forall n0 : nat,
(n0 >= N)%nat -> R_dist (h n0) 0 < eps
n:nat
y:R
nN:(N <= n)%nat
dy:Boule c r y
Rabs
  (sum_f_R0 (tg_alt (fun i : nat => f i y)) n - g y) <=
f n y
f:nat -> R -> R
g:R -> R
h:nat -> R
c:R
r:posreal
decr:forall x : R,
Boule c r x ->
Un_decreasing (fun n0 : nat => f n0 x)
to0:forall x : R,
Boule c r x -> Un_cv (fun n0 : nat => f n0 x) 0
to_g:forall x : R,
Boule c r x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => f i x)))
  (g x)
bound:forall (x : R) (n0 : nat),
Boule c r x -> f n0 x <= h n0
bound0:Un_cv h 0
eps:R
ep:0 < eps
ep':0 < eps / 2
N:nat
Pn:forall n0 : nat,
(n0 >= N)%nat -> R_dist (h n0) 0 < eps
n:nat
y:R
nN:(N <= n)%nat
dy:Boule c r y
f n y < eps
 solve[apply (Alt_first_term_bound (fun i => f i y) (g y) n n); auto].f:nat -> R -> R
g:R -> R
h:nat -> R
c:R
r:posreal
decr:forall x : R,
Boule c r x ->
Un_decreasing (fun n0 : nat => f n0 x)
to0:forall x : R,
Boule c r x -> Un_cv (fun n0 : nat => f n0 x) 0
to_g:forall x : R,
Boule c r x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => f i x)))
  (g x)
bound:forall (x : R) (n0 : nat),
Boule c r x -> f n0 x <= h n0
bound0:Un_cv h 0
eps:R
ep:0 < eps
ep':0 < eps / 2
N:nat
Pn:forall n0 : nat,
(n0 >= N)%nat -> R_dist (h n0) 0 < eps
n:nat
y:R
nN:(N <= n)%nat
dy:Boule c r y
f n y < eps
apply Rle_lt_trans with (h n).f:nat -> R -> R
g:R -> R
h:nat -> R
c:R
r:posreal
decr:forall x : R,
Boule c r x ->
Un_decreasing (fun n0 : nat => f n0 x)
to0:forall x : R,
Boule c r x -> Un_cv (fun n0 : nat => f n0 x) 0
to_g:forall x : R,
Boule c r x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => f i x)))
  (g x)
bound:forall (x : R) (n0 : nat),
Boule c r x -> f n0 x <= h n0
bound0:Un_cv h 0
eps:R
ep:0 < eps
ep':0 < eps / 2
N:nat
Pn:forall n0 : nat,
(n0 >= N)%nat -> R_dist (h n0) 0 < eps
n:nat
y:R
nN:(N <= n)%nat
dy:Boule c r y
f n y <= h n
f:nat -> R -> R
g:R -> R
h:nat -> R
c:R
r:posreal
decr:forall x : R,
Boule c r x ->
Un_decreasing (fun n0 : nat => f n0 x)
to0:forall x : R,
Boule c r x -> Un_cv (fun n0 : nat => f n0 x) 0
to_g:forall x : R,
Boule c r x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => f i x)))
  (g x)
bound:forall (x : R) (n0 : nat),
Boule c r x -> f n0 x <= h n0
bound0:Un_cv h 0
eps:R
ep:0 < eps
ep':0 < eps / 2
N:nat
Pn:forall n0 : nat,
(n0 >= N)%nat -> R_dist (h n0) 0 < eps
n:nat
y:R
nN:(N <= n)%nat
dy:Boule c r y
h n < eps
 apply bound; assumption.f:nat -> R -> R
g:R -> R
h:nat -> R
c:R
r:posreal
decr:forall x : R,
Boule c r x ->
Un_decreasing (fun n0 : nat => f n0 x)
to0:forall x : R,
Boule c r x -> Un_cv (fun n0 : nat => f n0 x) 0
to_g:forall x : R,
Boule c r x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => f i x)))
  (g x)
bound:forall (x : R) (n0 : nat),
Boule c r x -> f n0 x <= h n0
bound0:Un_cv h 0
eps:R
ep:0 < eps
ep':0 < eps / 2
N:nat
Pn:forall n0 : nat,
(n0 >= N)%nat -> R_dist (h n0) 0 < eps
n:nat
y:R
nN:(N <= n)%nat
dy:Boule c r y
h n < eps
clear - nN Pn.h:nat -> R
eps:R
N:nat
Pn:forall n0 : nat,
(n0 >= N)%nat -> R_dist (h n0) 0 < eps
n:nat
nN:(N <= n)%nat
h n < eps
generalize (Pn _ nN); unfold R_dist; rewrite Rminus_0_r; intros t.h:nat -> R
eps:R
N:nat
Pn:forall n0 : nat,
(n0 >= N)%nat -> R_dist (h n0) 0 < eps
n:nat
nN:(N <= n)%nat
t:Rabs (h n) < eps
h n < eps
apply Rabs_def2 in t; tauto.
Qed.

(* The following lemmas are general purpose lemmas about squares. 
  They do not belong here *)

Lemma pow2_ge_0 : forall x, 0 <= x ^ 2.forall x : R, 0 <= x ^ 2
Proof.forall x : R, 0 <= x ^ 2
intros x; destruct (Rle_lt_dec 0 x).x:R
r:0 <= x
0 <= x ^ 2
x:R
r:x < 0
0 <= x ^ 2
 replace (x ^ 2) with (x * x) by field.x:R
r:0 <= x
0 <= x * x
x:R
r:x < 0
0 <= x ^ 2
 apply Rmult_le_pos; assumption.x:R
r:x < 0
0 <= x ^ 2
 replace (x ^ 2) with ((-x) * (-x)) by field.x:R
r:x < 0
0 <= - x * - x
apply Rmult_le_pos; lra.
Qed.

Lemma pow2_abs : forall x,  Rabs x ^ 2 = x ^ 2.forall x : R, Rabs x ^ 2 = x ^ 2
Proof.forall x : R, Rabs x ^ 2 = x ^ 2
intros x; destruct (Rle_lt_dec 0 x).x:R
r:0 <= x
Rabs x ^ 2 = x ^ 2
x:R
r:x < 0
Rabs x ^ 2 = x ^ 2
 rewrite Rabs_pos_eq;[field | assumption].x:R
r:x < 0
Rabs x ^ 2 = x ^ 2
rewrite <- Rabs_Ropp, Rabs_pos_eq;[field | lra].
Qed.

Properties of tangent

Lemma derivable_pt_tan : forall x, -PI/2 < x < PI/2 -> derivable_pt tan x.forall x : R,
- PI / 2 < x < PI / 2 -> derivable_pt tan x
Proof.forall x : R,
- PI / 2 < x < PI / 2 -> derivable_pt tan x
intros x xint.x:R
xint:- PI / 2 < x < PI / 2
derivable_pt tan x
 unfold derivable_pt, tan.x:R
xint:- PI / 2 < x < PI / 2
{l : R |
derivable_pt_abs (fun x0 : R => sin x0 / cos x0) x l} 
 apply derivable_pt_div ; [reg | reg | ].x:R
xint:- PI / 2 < x < PI / 2
cos x <> 0
 apply Rgt_not_eq.x:R
xint:- PI / 2 < x < PI / 2
cos x > 0
 unfold Rgt ; apply cos_gt_0;
  [unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse; fold (-PI/2) |];tauto.
Qed.

Lemma derive_pt_tan : forall (x:R),
 forall (Pr1: -PI/2 < x < PI/2),
 derive_pt tan x (derivable_pt_tan x Pr1) = 1 + (tan x)^2.forall (x : R) (Pr1 : - PI / 2 < x < PI / 2),
derive_pt tan x (derivable_pt_tan x Pr1) =
1 + tan x ^ 2
Proof.forall (x : R) (Pr1 : - PI / 2 < x < PI / 2),
derive_pt tan x (derivable_pt_tan x Pr1) =
1 + tan x ^ 2
intros x pr.x:R
pr:- PI / 2 < x < PI / 2
derive_pt tan x (derivable_pt_tan x pr) =
1 + tan x ^ 2
assert (cos x <> 0).x:R
pr:- PI / 2 < x < PI / 2
cos x <> 0
x:R
pr:- PI / 2 < x < PI / 2
H:cos x <> 0
derive_pt tan x (derivable_pt_tan x pr) =
1 + tan x ^ 2
 apply Rgt_not_eq, cos_gt_0; rewrite <- ?Ropp_div; tauto.x:R
pr:- PI / 2 < x < PI / 2
H:cos x <> 0
derive_pt tan x (derivable_pt_tan x pr) =
1 + tan x ^ 2
unfold tan; reg; unfold pow, Rsqr; field; assumption.
Qed.

Proof that tangent is a bijection

(* to be removed? *)

Lemma derive_increasing_interv :
  forall (a b:R) (f:R -> R),
    a < b ->
    forall (pr:forall x, a < x < b -> derivable_pt f x),
    (forall t:R, forall (t_encad : a < t < b), 0 < derive_pt f t (pr t t_encad)) ->
    forall x y:R, a < x < b -> a < y < b -> x < y -> f x < f y.forall (a b : R) (f : R -> R),
a < b ->
forall
  pr : forall x : R, a < x < b -> derivable_pt f x,
(forall (t : R) (t_encad : a < t < b),
 0 < derive_pt f t (pr t t_encad)) ->
forall x y : R,
a < x < b -> a < y < b -> x < y -> f x < f y
Proof.forall (a b : R) (f : R -> R),
a < b ->
forall
  pr : forall x : R, a < x < b -> derivable_pt f x,
(forall (t : R) (t_encad : a < t < b),
 0 < derive_pt f t (pr t t_encad)) ->
forall x y : R,
a < x < b -> a < y < b -> x < y -> f x < f y
intros a b f a_lt_b pr Df_gt_0 x y x_encad y_encad x_lt_y.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
f x < f y
 assert (derivable_id_interv : forall c : R, x < c < y -> derivable_pt id c).a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
forall c : R, x < c < y -> derivable_pt id c
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
f x < f y
  intros ; apply derivable_pt_id.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
f x < f y
 assert (derivable_f_interv :  forall c : R, x < c < y -> derivable_pt f c).a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
forall c : R, x < c < y -> derivable_pt f c
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f x < f y
  intros c c_encad.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
c:R
c_encad:x < c < y
derivable_pt f c
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f x < f y apply pr.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
c:R
c_encad:x < c < y
a < c < b
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f x < f y split.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
c:R
c_encad:x < c < y
a < c
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
c:R
c_encad:x < c < y
c < b
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f x < f y
  apply Rlt_trans with (r2:=x) ; [exact (proj1 x_encad) | exact (proj1 c_encad)].a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
c:R
c_encad:x < c < y
c < b
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f x < f y
  apply Rlt_trans with (r2:=y) ; [exact (proj2 c_encad) | exact (proj2 y_encad)].a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f x < f y
 assert (f_cont_interv : forall c : R, x <= c <= y -> continuity_pt f c).a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
forall c : R, x <= c <= y -> continuity_pt f c
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f_cont_interv:forall c : R,
x <= c <= y -> continuity_pt f c
f x < f y
  intros c c_encad; apply derivable_continuous_pt ; apply pr.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
c:R
c_encad:x <= c <= y
a < c < b
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f_cont_interv:forall c : R,
x <= c <= y -> continuity_pt f c
f x < f y split.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
c:R
c_encad:x <= c <= y
a < c
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
c:R
c_encad:x <= c <= y
c < b
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f_cont_interv:forall c : R,
x <= c <= y -> continuity_pt f c
f x < f y
  apply Rlt_le_trans with (r2:=x) ; [exact (proj1 x_encad) | exact (proj1 c_encad)].a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
c:R
c_encad:x <= c <= y
c < b
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f_cont_interv:forall c : R,
x <= c <= y -> continuity_pt f c
f x < f y
  apply Rle_lt_trans with (r2:=y) ; [ exact (proj2 c_encad) | exact (proj2 y_encad)].a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f_cont_interv:forall c : R,
x <= c <= y -> continuity_pt f c
f x < f y
 assert (id_cont_interv : forall c : R, x <= c <= y -> continuity_pt id c).a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f_cont_interv:forall c : R,
x <= c <= y -> continuity_pt f c
forall c : R, x <= c <= y -> continuity_pt id c
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f_cont_interv:forall c : R,
x <= c <= y -> continuity_pt f c
id_cont_interv:forall c : R,
x <= c <= y -> continuity_pt id c
f x < f y
  intros ; apply derivable_continuous_pt ; apply derivable_pt_id.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f_cont_interv:forall c : R,
x <= c <= y -> continuity_pt f c
id_cont_interv:forall c : R,
x <= c <= y -> continuity_pt id c
f x < f y 
 elim (MVT f id x y derivable_f_interv derivable_id_interv x_lt_y f_cont_interv id_cont_interv).a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c : R,
x < c < y -> derivable_pt id c
derivable_f_interv:forall c : R,
x < c < y -> derivable_pt f c
f_cont_interv:forall c : R,
x <= c <= y -> continuity_pt f c
id_cont_interv:forall c : R,
x <= c <= y -> continuity_pt id c
forall x0 : R,
(exists P : x < x0 < y,
   (id y - id x) *
   derive_pt f x0 (derivable_f_interv x0 P) =
   (f y - f x) *
   derive_pt id x0 (derivable_id_interv x0 P)) ->
f x < f y
  intros c Temp ; elim Temp ; clear Temp ; intros Pr eq.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(id y - id x) *
derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
f x < f y
   replace (id y - id x) with (y - x) in eq by intuition.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
f x < f y
   replace (derive_pt id c (derivable_id_interv c Pr)) with 1 in eq.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
   assert (Hyp : f y - f x > 0).a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
f y - f x > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
    rewrite Rmult_1_r in eq.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
f y - f x > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr) rewrite <- eq.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
(y - x) * derive_pt f c (derivable_f_interv c Pr) > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
    apply Rmult_gt_0_compat.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
y - x > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
derive_pt f c (derivable_f_interv c Pr) > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
    apply Rgt_minus ; assumption.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
derive_pt f c (derivable_f_interv c Pr) > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
    assert (c_encad2 : a <= c < b).a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
a <= c < b
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
derive_pt f c (derivable_f_interv c Pr) > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
     split.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
a <= c
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c < b
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
derive_pt f c (derivable_f_interv c Pr) > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
     apply Rlt_le ; apply Rlt_trans with (r2:=x) ; [exact (proj1 x_encad) | exact (proj1 Pr)].a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c < b
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
derive_pt f c (derivable_f_interv c Pr) > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
     apply Rle_lt_trans with (r2:=y) ; [apply Rlt_le ; exact (proj2 Pr) | exact (proj2 y_encad)].a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
derive_pt f c (derivable_f_interv c Pr) > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
     assert (c_encad : a < c < b).a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
a < c < b
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
c_encad:a < c < b
derive_pt f c (derivable_f_interv c Pr) > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
      split.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
a < c
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
c < b
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
c_encad:a < c < b
derive_pt f c (derivable_f_interv c Pr) > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
      apply Rlt_trans with (r2:=x) ; [exact (proj1 x_encad) | exact (proj1 Pr)].a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
c < b
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
c_encad:a < c < b
derive_pt f c (derivable_f_interv c Pr) > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
      apply Rle_lt_trans with (r2:=y) ; [apply Rlt_le ; exact (proj2 Pr) | exact (proj2 y_encad)].a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
c_encad:a < c < b
derive_pt f c (derivable_f_interv c Pr) > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
     assert (Temp := Df_gt_0 c c_encad).a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
c_encad:a < c < b
Temp:0 < derive_pt f c (pr c c_encad)
derive_pt f c (derivable_f_interv c Pr) > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
     assert (Temp2 := pr_nu f c (derivable_f_interv c Pr) (pr c c_encad)).a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
f y - f x
c_encad2:a <= c < b
c_encad:a < c < b
Temp:0 < derive_pt f c (pr c c_encad)
Temp2:derive_pt f c (derivable_f_interv c Pr) =
derive_pt f c (pr c c_encad)
derive_pt f c (derivable_f_interv c Pr) > 0
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
     rewrite Temp2 ; apply Temp.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) * 1
Hyp:f y - f x > 0
f x < f y
a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
   apply Rminus_gt ; exact Hyp.a, b:R
f:R -> R
a_lt_b:a < b
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
Df_gt_0:forall (t : R) (t_encad : a < t < b),
0 < derive_pt f t (pr t t_encad)
x, y:R
x_encad:a < x < b
y_encad:a < y < b
x_lt_y:x < y
derivable_id_interv:forall c0 : R,
x < c0 < y -> derivable_pt id c0
derivable_f_interv:forall c0 : R,
x < c0 < y -> derivable_pt f c0
f_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt f c0
id_cont_interv:forall c0 : R,
x <= c0 <= y -> continuity_pt id c0
c:R
Pr:x < c < y
eq:(y - x) * derive_pt f c (derivable_f_interv c Pr) =
(f y - f x) *
derive_pt id c (derivable_id_interv c Pr)
1 = derive_pt id c (derivable_id_interv c Pr)
   symmetry ; rewrite derive_pt_eq ; apply derivable_pt_lim_id.
Qed.

(* begin hide *)
Lemma plus_Rsqr_gt_0 : forall x, 1 + x ^ 2 > 0.forall x : R, 1 + x ^ 2 > 0
Proof.forall x : R, 1 + x ^ 2 > 0
intro m.m:R
1 + m ^ 2 > 0 replace 0 with (0+0) by intuition.m:R
1 + m ^ 2 > 0 + 0
 apply Rplus_gt_ge_compat.m:R
1 > 0
m:R
m ^ 2 >= 0 intuition.m:R
m ^ 2 >= 0
 elim (total_order_T m 0) ; intro s'.m:R
s':{m < 0} + {m = 0}
m ^ 2 >= 0
m:R
s':m > 0
m ^ 2 >= 0 case s'.m:R
s':{m < 0} + {m = 0}
m < 0 -> m ^ 2 >= 0
m:R
s':{m < 0} + {m = 0}
m = 0 -> m ^ 2 >= 0
m:R
s':m > 0
m ^ 2 >= 0
 intros m_cond.m:R
s':{m < 0} + {m = 0}
m_cond:m < 0
m ^ 2 >= 0
m:R
s':{m < 0} + {m = 0}
m = 0 -> m ^ 2 >= 0
m:R
s':m > 0
m ^ 2 >= 0 replace 0 with (0*0) by intuition.m:R
s':{m < 0} + {m = 0}
m_cond:m < 0
m ^ 2 >= 0 * 0
m:R
s':{m < 0} + {m = 0}
m = 0 -> m ^ 2 >= 0
m:R
s':m > 0
m ^ 2 >= 0
  replace (m ^ 2) with ((-m)^2).m:R
s':{m < 0} + {m = 0}
m_cond:m < 0
(- m) ^ 2 >= 0 * 0
m:R
s':{m < 0} + {m = 0}
m_cond:m < 0
(- m) ^ 2 = m ^ 2
m:R
s':{m < 0} + {m = 0}
m = 0 -> m ^ 2 >= 0
m:R
s':m > 0
m ^ 2 >= 0
  apply Rle_ge ; apply Rmult_le_compat ; intuition ; apply Rlt_le ; rewrite Rmult_1_r ; intuition.m:R
s':{m < 0} + {m = 0}
m_cond:m < 0
(- m) ^ 2 = m ^ 2
m:R
s':{m < 0} + {m = 0}
m = 0 -> m ^ 2 >= 0
m:R
s':m > 0
m ^ 2 >= 0
  field.m:R
s':{m < 0} + {m = 0}
m = 0 -> m ^ 2 >= 0
m:R
s':m > 0
m ^ 2 >= 0
  intro H' ; rewrite H' ; right ; field.m:R
s':m > 0
m ^ 2 >= 0
   left.m:R
s':m > 0
m ^ 2 > 0 intuition.
Qed.
(* end hide *)

(* The following lemmas about PI should probably be in Rtrigo. *)

Lemma PI2_lower_bound :
  forall x, 0 < x < 2 -> 0 < cos x  -> x < PI/2.forall x : R, 0 < x < 2 -> 0 < cos x -> x < PI / 2
Proof.forall x : R, 0 < x < 2 -> 0 < cos x -> x < PI / 2
intros x [xp xlt2] cx.x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
x < PI / 2
destruct (Rtotal_order x (PI/2)) as [xltpi2 | [xeqpi2 | xgtpi2]].x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xltpi2:x < PI / 2
x < PI / 2
x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xeqpi2:x = PI / 2
x < PI / 2
x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xgtpi2:x > PI / 2
x < PI / 2
  assumption.x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xeqpi2:x = PI / 2
x < PI / 2
x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xgtpi2:x > PI / 2
x < PI / 2
 now case (Rgt_not_eq _ _ cx); rewrite xeqpi2, cos_PI2.x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xgtpi2:x > PI / 2
x < PI / 2
destruct (MVT_cor1 cos (PI/2) x derivable_cos xgtpi2) as
   [c [Pc [cint1 cint2]]].x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xgtpi2:x > PI / 2
c:R
Pc:cos x - cos (PI / 2) =
derive_pt cos c (derivable_cos c) * (x - PI / 2)
cint1:PI / 2 < c
cint2:c < x
x < PI / 2
revert Pc; rewrite cos_PI2, Rminus_0_r.x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xgtpi2:x > PI / 2
c:R
cint1:PI / 2 < c
cint2:c < x
cos x =
derive_pt cos c (derivable_cos c) * (x - PI / 2) ->
x < PI / 2 
rewrite <- (pr_nu cos c (derivable_pt_cos c)), derive_pt_cos.x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xgtpi2:x > PI / 2
c:R
cint1:PI / 2 < c
cint2:c < x
cos x = - sin c * (x - PI / 2) -> x < PI / 2
assert (0 < c < 2) by (split; assert (t := PI2_RGT_0); lra).x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xgtpi2:x > PI / 2
c:R
cint1:PI / 2 < c
cint2:c < x
H:0 < c < 2
cos x = - sin c * (x - PI / 2) -> x < PI / 2
assert (0 < sin c) by now apply sin_pos_tech.x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xgtpi2:x > PI / 2
c:R
cint1:PI / 2 < c
cint2:c < x
H:0 < c < 2
H0:0 < sin c
cos x = - sin c * (x - PI / 2) -> x < PI / 2
intros Pc.x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xgtpi2:x > PI / 2
c:R
cint1:PI / 2 < c
cint2:c < x
H:0 < c < 2
H0:0 < sin c
Pc:cos x = - sin c * (x - PI / 2)
x < PI / 2
case (Rlt_not_le _ _ cx).x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xgtpi2:x > PI / 2
c:R
cint1:PI / 2 < c
cint2:c < x
H:0 < c < 2
H0:0 < sin c
Pc:cos x = - sin c * (x - PI / 2)
cos x <= 0
rewrite <- (Rplus_0_l (cos x)), Pc, Ropp_mult_distr_l_reverse.x:R
xp:0 < x
xlt2:x < 2
cx:0 < cos x
xgtpi2:x > PI / 2
c:R
cint1:PI / 2 < c
cint2:c < x
H:0 < c < 2
H0:0 < sin c
Pc:cos x = - sin c * (x - PI / 2)
0 + - (sin c * (x - PI / 2)) <= 0
apply Rle_minus, Rmult_le_pos;[apply Rlt_le; assumption | lra ].
Qed.

Lemma PI2_3_2 : 3/2 < PI/2.3 / 2 < PI / 2
Proof.3 / 2 < PI / 2
apply PI2_lower_bound;[split; lra | ].0 < cos (3 / 2)
destruct (pre_cos_bound (3/2) 1) as [t _]; [lra | lra | ].t:cos_approx (3 / 2) (2 * 1 + 1) <= cos (3 / 2)
0 < cos (3 / 2)
apply Rlt_le_trans with (2 := t); clear t.0 < cos_approx (3 / 2) (2 * 1 + 1)
unfold cos_approx; simpl; unfold cos_term.0 <
(-1) ^ 0 * ((3 / 2) ^ (2 * 0) / INR (fact (2 * 0))) +
(-1) ^ 1 * ((3 / 2) ^ (2 * 1) / INR (fact (2 * 1))) +
(-1) ^ 2 * ((3 / 2) ^ (2 * 2) / INR (fact (2 * 2))) +
(-1) ^ 3 * ((3 / 2) ^ (2 * 3) / INR (fact (2 * 3)))
rewrite !INR_IZR_INZ.0 <
(-1) ^ 0 *
((3 / 2) ^ (2 * 0) / IZR (Z.of_nat (fact (2 * 0)))) +
(-1) ^ 1 *
((3 / 2) ^ (2 * 1) / IZR (Z.of_nat (fact (2 * 1)))) +
(-1) ^ 2 *
((3 / 2) ^ (2 * 2) / IZR (Z.of_nat (fact (2 * 2)))) +
(-1) ^ 3 *
((3 / 2) ^ (2 * 3) / IZR (Z.of_nat (fact (2 * 3))))
simpl.0 <
1 * (1 / 1) + -1 * 1 * (3 / 2 * (3 / 2 * 1) / 2) +
-1 * (-1 * 1) *
(3 / 2 * (3 / 2 * (3 / 2 * (3 / 2 * 1))) / 24) +
-1 * (-1 * (-1 * 1)) *
(3 / 2 *
 (3 / 2 * (3 / 2 * (3 / 2 * (3 / 2 * (3 / 2 * 1))))) /
 720)
field_simplify.0 < 9925632 / 141557760
apply Rdiv_lt_0_compat ; now apply IZR_lt.
Qed.

Lemma PI2_1 : 1 < PI/2.1 < PI / 2
Proof.1 < PI / 2 assert (t := PI2_3_2); lra. Qed.

Lemma tan_increasing :
  forall x y:R,
    -PI/2 < x ->
    x < y -> 
    y < PI/2 -> tan x < tan y.forall x y : R,
- PI / 2 < x -> x < y -> y < PI / 2 -> tan x < tan y
Proof.forall x y : R,
- PI / 2 < x -> x < y -> y < PI / 2 -> tan x < tan y
intros x y Z_le_x x_lt_y y_le_1.x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
tan x < tan y
 assert (x_encad : -PI/2 < x < PI/2).x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
- PI / 2 < x < PI / 2
x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
x_encad:- PI / 2 < x < PI / 2
tan x < tan y
  split ; [assumption | apply Rlt_trans with (r2:=y) ; assumption].x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
x_encad:- PI / 2 < x < PI / 2
tan x < tan y
 assert (y_encad : -PI/2 < y < PI/2).x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
x_encad:- PI / 2 < x < PI / 2
- PI / 2 < y < PI / 2
x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
x_encad:- PI / 2 < x < PI / 2
y_encad:- PI / 2 < y < PI / 2
tan x < tan y
  split ; [apply Rlt_trans with (r2:=x) ; intuition | intuition ].x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
x_encad:- PI / 2 < x < PI / 2
y_encad:- PI / 2 < y < PI / 2
tan x < tan y
 assert (local_derivable_pt_tan : forall x : R, -PI/2 < x < PI/2 ->
          derivable_pt tan x).x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
x_encad:- PI / 2 < x < PI / 2
y_encad:- PI / 2 < y < PI / 2
forall x0 : R,
- PI / 2 < x0 < PI / 2 -> derivable_pt tan x0
x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
x_encad:- PI / 2 < x < PI / 2
y_encad:- PI / 2 < y < PI / 2
local_derivable_pt_tan:forall x0 : R,
- PI / 2 < x0 < PI / 2 ->
derivable_pt tan x0
tan x < tan y
  intros ; apply derivable_pt_tan ; intuition.x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
x_encad:- PI / 2 < x < PI / 2
y_encad:- PI / 2 < y < PI / 2
local_derivable_pt_tan:forall x0 : R,
- PI / 2 < x0 < PI / 2 ->
derivable_pt tan x0
tan x < tan y
 apply derive_increasing_interv with (a:=-PI/2) (b:=PI/2) (pr:=local_derivable_pt_tan) ; intuition.x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
local_derivable_pt_tan:forall x0 : R,
- PI / 2 < x0 < PI / 2 ->
derivable_pt tan x0
H:- PI / 2 < x
H0:x < PI / 2
H1:- PI / 2 < y
H2:y < PI / 2
- PI / 2 < PI / 2
x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
local_derivable_pt_tan:forall x0 : R,
- PI / 2 < x0 < PI / 2 ->
derivable_pt tan x0
H:- PI / 2 < x
H0:x < PI / 2
H1:- PI / 2 < y
H2:y < PI / 2
t:R
t_encad:- PI / 2 < t < PI / 2
0 < derive_pt tan t (local_derivable_pt_tan t t_encad)
 lra.x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
local_derivable_pt_tan:forall x0 : R,
- PI / 2 < x0 < PI / 2 ->
derivable_pt tan x0
H:- PI / 2 < x
H0:x < PI / 2
H1:- PI / 2 < y
H2:y < PI / 2
t:R
t_encad:- PI / 2 < t < PI / 2
0 < derive_pt tan t (local_derivable_pt_tan t t_encad)
 assert (Temp := pr_nu tan t (derivable_pt_tan t t_encad) (local_derivable_pt_tan t t_encad)) ;
 rewrite <- Temp ; clear Temp.x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
local_derivable_pt_tan:forall x0 : R,
- PI / 2 < x0 < PI / 2 ->
derivable_pt tan x0
H:- PI / 2 < x
H0:x < PI / 2
H1:- PI / 2 < y
H2:y < PI / 2
t:R
t_encad:- PI / 2 < t < PI / 2
0 < derive_pt tan t (derivable_pt_tan t t_encad)
 assert (Temp := derive_pt_tan t t_encad) ; rewrite Temp ; clear Temp.x, y:R
Z_le_x:- PI / 2 < x
x_lt_y:x < y
y_le_1:y < PI / 2
local_derivable_pt_tan:forall x0 : R,
- PI / 2 < x0 < PI / 2 ->
derivable_pt tan x0
H:- PI / 2 < x
H0:x < PI / 2
H1:- PI / 2 < y
H2:y < PI / 2
t:R
t_encad:- PI / 2 < t < PI / 2
0 < 1 + tan t ^ 2
 apply plus_Rsqr_gt_0.
Qed.

Lemma tan_is_inj : forall x y, -PI/2 < x < PI/2 -> -PI/2 < y < PI/2 ->
   tan x = tan y -> x = y.forall x y : R,
- PI / 2 < x < PI / 2 ->
- PI / 2 < y < PI / 2 -> tan x = tan y -> x = y
Proof.forall x y : R,
- PI / 2 < x < PI / 2 ->
- PI / 2 < y < PI / 2 -> tan x = tan y -> x = y
  intros a b a_encad b_encad fa_eq_fb.a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
a = b
  case(total_order_T a b).a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
{a < b} + {a = b} -> a = b
a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
a > b -> a = b
  intro s ; case s ; clear s.a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
a < b -> a = b
a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
a = b -> a = b
a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
a > b -> a = b
  intro Hf.a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
Hf:a < b
a = b
a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
a = b -> a = b
a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
a > b -> a = b
  assert (Hfalse := tan_increasing a b (proj1 a_encad) Hf (proj2 b_encad)).a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
Hf:a < b
Hfalse:tan a < tan b
a = b
a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
a = b -> a = b
a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
a > b -> a = b
  case (Rlt_not_eq (tan a) (tan b)) ; assumption.a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
a = b -> a = b
a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
a > b -> a = b
  intuition.a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
a > b -> a = b
  intro Hf.a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
Hf:a > b
a = b assert (Hfalse := tan_increasing b a (proj1 b_encad) Hf (proj2 a_encad)).a, b:R
a_encad:- PI / 2 < a < PI / 2
b_encad:- PI / 2 < b < PI / 2
fa_eq_fb:tan a = tan b
Hf:a > b
Hfalse:tan b < tan a
a = b
  case (Rlt_not_eq (tan b) (tan a)) ; [|symmetry] ; assumption.
Qed.

Lemma exists_atan_in_frame :
 forall lb ub y, lb < ub -> -PI/2 < lb -> ub < PI/2 ->
 tan lb < y < tan ub -> {x | lb < x < ub /\ tan x = y}.forall lb ub y : R,
lb < ub ->
- PI / 2 < lb ->
ub < PI / 2 ->
tan lb < y < tan ub ->
{x : R | lb < x < ub /\ tan x = y}
Proof.forall lb ub y : R,
lb < ub ->
- PI / 2 < lb ->
ub < PI / 2 ->
tan lb < y < tan ub ->
{x : R | lb < x < ub /\ tan x = y}
intros lb ub y lb_lt_ub lb_cond ub_cond y_encad.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
{x : R | lb < x < ub /\ tan x = y}
 case y_encad ; intros y_encad1 y_encad2.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
{x : R | lb < x < ub /\ tan x = y}
     assert (f_cont : forall a : R, lb <= a <= ub -> continuity_pt tan a).lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
forall a : R, lb <= a <= ub -> continuity_pt tan a
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
{x : R | lb < x < ub /\ tan x = y}
      intros a a_encad.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
a:R
a_encad:lb <= a <= ub
continuity_pt tan a
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
{x : R | lb < x < ub /\ tan x = y} apply derivable_continuous_pt ; apply derivable_pt_tan.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
a:R
a_encad:lb <= a <= ub
- PI / 2 < a < PI / 2
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
{x : R | lb < x < ub /\ tan x = y}
      split.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
a:R
a_encad:lb <= a <= ub
- PI / 2 < a
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
a:R
a_encad:lb <= a <= ub
a < PI / 2
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
{x : R | lb < x < ub /\ tan x = y} apply Rlt_le_trans with (r2:=lb) ; intuition.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
a:R
a_encad:lb <= a <= ub
a < PI / 2
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
{x : R | lb < x < ub /\ tan x = y}
      apply Rle_lt_trans with (r2:=ub) ; intuition.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
{x : R | lb < x < ub /\ tan x = y}
     assert (Cont : forall a : R, lb <= a <= ub -> continuity_pt (fun x => tan x - y) a).lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
{x : R | lb < x < ub /\ tan x = y}
      intros a a_encad.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a0 : R,
lb <= a0 <= ub -> continuity_pt tan a0
a:R
a_encad:lb <= a <= ub
continuity_pt (fun x : R => tan x - y) a
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
{x : R | lb < x < ub /\ tan x = y} unfold continuity_pt, continue_in, limit1_in, limit_in ; simpl ; unfold R_dist.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a0 : R,
lb <= a0 <= ub -> continuity_pt tan a0
a:R
a_encad:lb <= a <= ub
forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D_x no_cond a x /\ Rabs (x - a) < alp ->
   Rabs (tan x - y - (tan a - y)) < eps)
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
{x : R | lb < x < ub /\ tan x = y}
      intros eps eps_pos.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a0 : R,
lb <= a0 <= ub -> continuity_pt tan a0
a:R
a_encad:lb <= a <= ub
eps:R
eps_pos:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D_x no_cond a x /\ Rabs (x - a) < alp ->
   Rabs (tan x - y - (tan a - y)) < eps)
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
{x : R | lb < x < ub /\ tan x = y} elim (f_cont a a_encad eps eps_pos).lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a0 : R,
lb <= a0 <= ub -> continuity_pt tan a0
a:R
a_encad:lb <= a <= ub
eps:R
eps_pos:eps > 0
forall x : R,
x > 0 /\
(forall x0 : Base R_met,
 D_x no_cond a x0 /\ dist R_met x0 a < x ->
 dist R_met (tan x0) (tan a) < eps) ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond a x0 /\ Rabs (x0 - a) < alp ->
   Rabs (tan x0 - y - (tan a - y)) < eps)
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
{x : R | lb < x < ub /\ tan x = y}
      intros alpha alpha_pos.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a0 : R,
lb <= a0 <= ub -> continuity_pt tan a0
a:R
a_encad:lb <= a <= ub
eps:R
eps_pos:eps > 0
alpha:R
alpha_pos:alpha > 0 /\
(forall x : Base R_met,
 D_x no_cond a x /\ dist R_met x a < alpha ->
 dist R_met (tan x) (tan a) < eps)
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D_x no_cond a x /\ Rabs (x - a) < alp ->
   Rabs (tan x - y - (tan a - y)) < eps)
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
{x : R | lb < x < ub /\ tan x = y} destruct alpha_pos as (alpha_pos,Temp).lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a0 : R,
lb <= a0 <= ub -> continuity_pt tan a0
a:R
a_encad:lb <= a <= ub
eps:R
eps_pos:eps > 0
alpha:R
alpha_pos:alpha > 0
Temp:forall x : Base R_met,
D_x no_cond a x /\ dist R_met x a < alpha ->
dist R_met (tan x) (tan a) < eps
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D_x no_cond a x /\ Rabs (x - a) < alp ->
   Rabs (tan x - y - (tan a - y)) < eps)
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
{x : R | lb < x < ub /\ tan x = y}
      exists alpha.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a0 : R,
lb <= a0 <= ub -> continuity_pt tan a0
a:R
a_encad:lb <= a <= ub
eps:R
eps_pos:eps > 0
alpha:R
alpha_pos:alpha > 0
Temp:forall x : Base R_met,
D_x no_cond a x /\ dist R_met x a < alpha ->
dist R_met (tan x) (tan a) < eps
alpha > 0 /\
(forall x : R,
 D_x no_cond a x /\ Rabs (x - a) < alpha ->
 Rabs (tan x - y - (tan a - y)) < eps)
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
{x : R | lb < x < ub /\ tan x = y} split.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a0 : R,
lb <= a0 <= ub -> continuity_pt tan a0
a:R
a_encad:lb <= a <= ub
eps:R
eps_pos:eps > 0
alpha:R
alpha_pos:alpha > 0
Temp:forall x : Base R_met,
D_x no_cond a x /\ dist R_met x a < alpha ->
dist R_met (tan x) (tan a) < eps
alpha > 0
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a0 : R,
lb <= a0 <= ub -> continuity_pt tan a0
a:R
a_encad:lb <= a <= ub
eps:R
eps_pos:eps > 0
alpha:R
alpha_pos:alpha > 0
Temp:forall x : Base R_met,
D_x no_cond a x /\ dist R_met x a < alpha ->
dist R_met (tan x) (tan a) < eps
forall x : R,
D_x no_cond a x /\ Rabs (x - a) < alpha ->
Rabs (tan x - y - (tan a - y)) < eps
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
{x : R | lb < x < ub /\ tan x = y}
      assumption.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a0 : R,
lb <= a0 <= ub -> continuity_pt tan a0
a:R
a_encad:lb <= a <= ub
eps:R
eps_pos:eps > 0
alpha:R
alpha_pos:alpha > 0
Temp:forall x : Base R_met,
D_x no_cond a x /\ dist R_met x a < alpha ->
dist R_met (tan x) (tan a) < eps
forall x : R,
D_x no_cond a x /\ Rabs (x - a) < alpha ->
Rabs (tan x - y - (tan a - y)) < eps
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
{x : R | lb < x < ub /\ tan x = y} intros x  x_cond.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a0 : R,
lb <= a0 <= ub -> continuity_pt tan a0
a:R
a_encad:lb <= a <= ub
eps:R
eps_pos:eps > 0
alpha:R
alpha_pos:alpha > 0
Temp:forall x0 : Base R_met,
D_x no_cond a x0 /\ dist R_met x0 a < alpha ->
dist R_met (tan x0) (tan a) < eps
x:R
x_cond:D_x no_cond a x /\ Rabs (x - a) < alpha
Rabs (tan x - y - (tan a - y)) < eps
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
{x : R | lb < x < ub /\ tan x = y}
      replace (tan x - y - (tan a - y)) with (tan x - tan a) by field.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a0 : R,
lb <= a0 <= ub -> continuity_pt tan a0
a:R
a_encad:lb <= a <= ub
eps:R
eps_pos:eps > 0
alpha:R
alpha_pos:alpha > 0
Temp:forall x0 : Base R_met,
D_x no_cond a x0 /\ dist R_met x0 a < alpha ->
dist R_met (tan x0) (tan a) < eps
x:R
x_cond:D_x no_cond a x /\ Rabs (x - a) < alpha
Rabs (tan x - tan a) < eps
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
{x : R | lb < x < ub /\ tan x = y}
      exact (Temp x x_cond).lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
{x : R | lb < x < ub /\ tan x = y}
     assert (H1 : (fun x : R => tan x - y) lb < 0).lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
tan lb - y < 0
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
H1:(fun x : R => tan x - y) lb < 0
{x : R | lb < x < ub /\ tan x = y}
       apply Rlt_minus.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
tan lb < y
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
H1:(fun x : R => tan x - y) lb < 0
{x : R | lb < x < ub /\ tan x = y} assumption.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
H1:(fun x : R => tan x - y) lb < 0
{x : R | lb < x < ub /\ tan x = y}
      assert (H2 : 0 < (fun x : R => tan x - y) ub).lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
H1:(fun x : R => tan x - y) lb < 0
0 < tan ub - y
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
H1:(fun x : R => tan x - y) lb < 0
H2:0 < (fun x : R => tan x - y) ub
{x : R | lb < x < ub /\ tan x = y}
       apply Rgt_minus.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
H1:(fun x : R => tan x - y) lb < 0
tan ub > y
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
H1:(fun x : R => tan x - y) lb < 0
H2:0 < (fun x : R => tan x - y) ub
{x : R | lb < x < ub /\ tan x = y} assumption.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x : R => tan x - y) a
H1:(fun x : R => tan x - y) lb < 0
H2:0 < (fun x : R => tan x - y) ub
{x : R | lb < x < ub /\ tan x = y}
     destruct (IVT_interv (fun x => tan x - y) lb ub Cont lb_lt_ub H1 H2) as (x,Hx).lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:(fun x0 : R => tan x0 - y) lb < 0
H2:0 < (fun x0 : R => tan x0 - y) ub
x:R
Hx:lb <= x <= ub /\ tan x - y = 0
{x0 : R | lb < x0 < ub /\ tan x0 = y}
     exists x.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:(fun x0 : R => tan x0 - y) lb < 0
H2:0 < (fun x0 : R => tan x0 - y) ub
x:R
Hx:lb <= x <= ub /\ tan x - y = 0
lb < x < ub /\ tan x = y
     destruct Hx as (Hyp,Result).lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad:tan lb < y < tan ub
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:(fun x0 : R => tan x0 - y) lb < 0
H2:0 < (fun x0 : R => tan x0 - y) ub
x:R
Hyp:lb <= x <= ub
Result:tan x - y = 0
lb < x < ub /\ tan x = y
     intuition.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
lb < x
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x < ub
     assert (Temp2 : x <> lb).lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x <> lb
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp2:x <> lb
lb < x
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x < ub
      intro Hfalse.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Hfalse:x = lb
False
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp2:x <> lb
lb < x
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x < ub rewrite Hfalse in Result.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan lb - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Hfalse:x = lb
False
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp2:x <> lb
lb < x
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x < ub
      assert (Temp2 : y <> tan lb).lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan lb - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Hfalse:x = lb
y <> tan lb
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan lb - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Hfalse:x = lb
Temp2:y <> tan lb
False
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp2:x <> lb
lb < x
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x < ub
       apply Rgt_not_eq ; assumption.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan lb - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Hfalse:x = lb
Temp2:y <> tan lb
False
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp2:x <> lb
lb < x
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x < ub
      clear - Temp2 Result.lb, y:R
Result:tan lb - y = 0
Temp2:y <> tan lb
False
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp2:x <> lb
lb < x
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x < ub apply Temp2.lb, y:R
Result:tan lb - y = 0
Temp2:y <> tan lb
y = tan lb
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp2:x <> lb
lb < x
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x < ub
      intuition.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp2:x <> lb
lb < x
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x < ub
     clear -Temp2 H3.lb, x:R
H3:lb <= x
Temp2:x <> lb
lb < x
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x < ub
      case H3 ; intuition.lb, x:R
H3:lb <= x
Temp2:x = lb -> False
H:lb = x
lb < x
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x < ub apply False_ind ; apply Temp2 ; symmetry ; assumption.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x < ub
      assert (Temp : x <> ub).lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
x <> ub
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp:x <> ub
x < ub
      intro Hfalse.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Hfalse:x = ub
False
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp:x <> ub
x < ub rewrite Hfalse in Result.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan ub - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Hfalse:x = ub
False
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp:x <> ub
x < ub
      assert (Temp2 : y <> tan ub).lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan ub - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Hfalse:x = ub
y <> tan ub
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan ub - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Hfalse:x = ub
Temp2:y <> tan ub
False
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp:x <> ub
x < ub
       apply Rlt_not_eq ; assumption.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan ub - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Hfalse:x = ub
Temp2:y <> tan ub
False
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp:x <> ub
x < ub
      clear - Temp2 Result.ub, y:R
Result:tan ub - y = 0
Temp2:y <> tan ub
False
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp:x <> ub
x < ub apply Temp2.ub, y:R
Result:tan ub - y = 0
Temp2:y <> tan ub
y = tan ub
lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp:x <> ub
x < ub
      intuition.lb, ub, y:R
lb_lt_ub:lb < ub
lb_cond:- PI / 2 < lb
ub_cond:ub < PI / 2
y_encad1:tan lb < y
y_encad2:y < tan ub
f_cont:forall a : R,
lb <= a <= ub -> continuity_pt tan a
Cont:forall a : R,
lb <= a <= ub ->
continuity_pt (fun x0 : R => tan x0 - y) a
H1:tan lb - y < 0
H2:0 < tan ub - y
x:R
Result:tan x - y = 0
H:tan lb < y
H0:y < tan ub
H3:lb <= x
H4:x <= ub
Temp:x <> ub
x < ub
      case H4 ; intuition.
Qed.

Definition of arctangent as the reciprocal function of tangent and proof of this status

Lemma tan_1_gt_1 : tan 1 > 1.tan 1 > 1
Proof.tan 1 > 1
assert (0 < cos 1) by (apply cos_gt_0; assert (t:=PI2_1); lra).H:0 < cos 1
tan 1 > 1
assert (t1 : cos 1 <= 1 - 1/2 + 1/24).H:0 < cos 1
cos 1 <= 1 - 1 / 2 + 1 / 24
H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
tan 1 > 1
 destruct (pre_cos_bound 1 0) as [_ t]; try lra; revert t.H:0 < cos 1
cos 1 <= cos_approx 1 (2 * (0 + 1)) ->
cos 1 <= 1 - 1 / 2 + 1 / 24
H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
tan 1 > 1
 unfold cos_approx, cos_term; simpl; intros t; apply Rle_trans with (1:=t).H:0 < cos 1
t:cos 1 <=
1 * (1 / 1) + -1 * 1 * (1 * (1 * 1) / (1 + 1)) +
-1 * (-1 * 1) *
(1 * (1 * (1 * (1 * 1))) /
 (... + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 +
  1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1))
1 * (1 / 1) + -1 * 1 * (1 * (1 * 1) / (1 + 1)) +
-1 * (-1 * 1) *
(1 * (1 * (1 * (1 * 1))) /
 (... + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 +
  1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1)) <=
1 - 1 / 2 + 1 / 24
H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
tan 1 > 1
 clear t; apply Req_le; field.H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
tan 1 > 1
assert (t2 : 1 - 1/6 <= sin 1).H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
1 - 1 / 6 <= sin 1
H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
t2:1 - 1 / 6 <= sin 1
tan 1 > 1
 destruct (pre_sin_bound 1 0) as [t _]; try lra; revert t.H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
sin_approx 1 (2 * 0 + 1) <= sin 1 ->
1 - 1 / 6 <= sin 1
H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
t2:1 - 1 / 6 <= sin 1
tan 1 > 1
 unfold sin_approx, sin_term; simpl; intros t; apply Rle_trans with (2:=t).H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
t:1 * (1 * 1 / 1) +
-1 * 1 *
(1 * (1 * (1 * 1)) / (1 + 1 + 1 + 1 + 1 + 1)) <=
sin 1
1 - 1 / 6 <=
1 * (1 * 1 / 1) +
-1 * 1 * (1 * (1 * (1 * 1)) / (1 + 1 + 1 + 1 + 1 + 1))
H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
t2:1 - 1 / 6 <= sin 1
tan 1 > 1
 clear t; apply Req_le; field.H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
t2:1 - 1 / 6 <= sin 1
tan 1 > 1
pattern 1 at 2; replace 1 with
  (cos 1 / cos 1) by (field; apply Rgt_not_eq; lra).H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
t2:1 - 1 / 6 <= sin 1
tan 1 > cos 1 / cos 1
apply Rlt_gt; apply (Rmult_lt_compat_r (/ cos 1) (cos 1) (sin 1)).H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
t2:1 - 1 / 6 <= sin 1
0 < / cos 1
H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
t2:1 - 1 / 6 <= sin 1
cos 1 < sin 1
 apply Rinv_0_lt_compat; assumption.H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
t2:1 - 1 / 6 <= sin 1
cos 1 < sin 1
apply Rle_lt_trans with (1 := t1); apply Rlt_le_trans with (2 := t2).H:0 < cos 1
t1:cos 1 <= 1 - 1 / 2 + 1 / 24
t2:1 - 1 / 6 <= sin 1
1 - 1 / 2 + 1 / 24 < 1 - 1 / 6
lra.
Qed.

Definition frame_tan y : {x | 0 < x < PI/2 /\ Rabs y < tan x}.y:R
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
Proof.y:R
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
destruct (total_order_T (Rabs y) 1) as [Hs|Hgt].y:R
Hs:{Rabs y < 1} + {Rabs y = 1}
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
y:R
Hgt:Rabs y > 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
 assert (yle1 : Rabs y <= 1) by (destruct Hs; lra).y:R
Hs:{Rabs y < 1} + {Rabs y = 1}
yle1:Rabs y <= 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
y:R
Hgt:Rabs y > 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
 clear Hs; exists 1; split;[split; [exact Rlt_0_1 | exact PI2_1] | ].y:R
yle1:Rabs y <= 1
Rabs y < tan 1
y:R
Hgt:Rabs y > 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
 apply Rle_lt_trans with (1 := yle1); exact tan_1_gt_1.y:R
Hgt:Rabs y > 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
assert (0 < / (Rabs y + 1)).y:R
Hgt:Rabs y > 1
0 < / (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
 apply Rinv_0_lt_compat; lra.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
set (u := /2 * / (Rabs y + 1)).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
assert (0 < u).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
0 < u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
 apply Rmult_lt_0_compat; [lra | assumption].y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
assert (vlt1 : / (Rabs y + 1) < 1).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
/ (Rabs y + 1) < 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
 apply Rmult_lt_reg_r with (Rabs y + 1).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
0 < Rabs y + 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
/ (Rabs y + 1) * (Rabs y + 1) < 1 * (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
  assert (t := Rabs_pos y); lra.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
/ (Rabs y + 1) * (Rabs y + 1) < 1 * (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
 rewrite Rinv_l; [rewrite Rmult_1_l | apply Rgt_not_eq]; lra.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
assert (vlt2 : u < 1).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
u < 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
 apply Rlt_trans with (/ (Rabs y + 1)).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
u < / (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
/ (Rabs y + 1) < 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
  rewrite double_var.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
u < / (Rabs y + 1) / 2 + / (Rabs y + 1) / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
/ (Rabs y + 1) < 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
  assert (t : forall x, 0 < x -> x < x + x) by (clear; intros; lra).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
t:forall x : R, 0 < x -> x < x + x
u < / (Rabs y + 1) / 2 + / (Rabs y + 1) / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
/ (Rabs y + 1) < 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
  unfold u; rewrite Rmult_comm; apply t.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
t:forall x : R, 0 < x -> x < x + x
0 < / (Rabs y + 1) / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
/ (Rabs y + 1) < 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
  unfold Rdiv; rewrite Rmult_comm; assumption.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
/ (Rabs y + 1) < 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
 assumption.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
assert(int : 0 < PI / 2 - u < PI / 2).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
0 < PI / 2 - u < PI / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
 split.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
0 < PI / 2 - u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
PI / 2 - u < PI / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
  assert (t := PI2_1); apply Rlt_Rminus, Rlt_trans with (2 := t); assumption.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
PI / 2 - u < PI / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
 assert (dumb : forall x y, 0 < y -> x - y < x) by (clear; intros; lra).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
dumb:forall x y0 : R, 0 < y0 -> x - y0 < x
PI / 2 - u < PI / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
 apply dumb; clear dumb; assumption.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
{x : R | 0 < x < PI / 2 /\ Rabs y < tan x}
exists (PI/2 - u).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
0 < PI / 2 - u < PI / 2 /\ Rabs y < tan (PI / 2 - u)
assert (tmp : forall x y, 0 < x -> y < 1 -> x * y < x).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
0 < PI / 2 - u < PI / 2 /\ Rabs y < tan (PI / 2 - u)
 clear; intros x y x0 y1; pattern x at 2; rewrite <- (Rmult_1_r x).x, y:R
x0:0 < x
y1:y < 1
x * y < x * 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
0 < PI / 2 - u < PI / 2 /\ Rabs y < tan (PI / 2 - u)
 apply Rmult_lt_compat_l; assumption.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
0 < PI / 2 - u < PI / 2 /\ Rabs y < tan (PI / 2 - u)
assert (0 < sin u).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
0 < sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
0 < PI / 2 - u < PI / 2 /\ Rabs y < tan (PI / 2 - u)
 apply sin_gt_0;[ assumption | ].y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
u < PI
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
0 < PI / 2 - u < PI / 2 /\ Rabs y < tan (PI / 2 - u)
 assert (t := PI2_Rlt_PI); assert (t' := PI2_1).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
t:PI / 2 < PI
t':1 < PI / 2
u < PI
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
0 < PI / 2 - u < PI / 2 /\ Rabs y < tan (PI / 2 - u)
 apply Rlt_trans with (2 := Rlt_trans _ _ _ t' t); assumption.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
0 < PI / 2 - u < PI / 2 /\ Rabs y < tan (PI / 2 - u)
split.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
0 < PI / 2 - u < PI / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
Rabs y < tan (PI / 2 - u)
 assumption.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
Rabs y < tan (PI / 2 - u)
 apply Rlt_trans with (/2 * / cos(PI / 2 - u)).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
Rabs y < / 2 * / cos (PI / 2 - u)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
 rewrite cos_shift.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
 assert (sin u < u).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
sin u < u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  assert (t1 : 0 <= u) by (apply Rlt_le; assumption).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
t1:0 <= u
sin u < u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  assert (t2 : u <= 4) by
   (apply Rle_trans with 1;[apply Rlt_le | lra]; assumption).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
t1:0 <= u
t2:u <= 4
sin u < u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  destruct (pre_sin_bound u 0 t1 t2) as [_ t].y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
t1:0 <= u
t2:u <= 4
t:sin u <= sin_approx u (2 * (0 + 1))
sin u < u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  apply Rle_lt_trans with (1 := t); clear t1 t2 t.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
sin_approx u (2 * (0 + 1)) < u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  unfold sin_approx; simpl; unfold sin_term; simpl ((-1) ^ 0);
  replace ((-1) ^ 2) with 1 by ring; simpl ((-1) ^ 1);
  rewrite !Rmult_1_r, !Rmult_1_l; simpl plus; simpl (INR (fact 1)).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u ^ 1 / 1 + -1 * (u ^ 3 / INR (fact 3)) +
u ^ 5 / INR (fact 5) < u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  rewrite <- (fun x => tech_pow_Rmult x 0), <- (fun x => tech_pow_Rmult x 2),
    <- (fun x => tech_pow_Rmult x 4).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u * u ^ 0 / 1 + -1 * (u * u ^ 2 / INR (fact 3)) +
u * u ^ 4 / INR (fact 5) < u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  rewrite (Rmult_comm (-1)); simpl ((/(Rabs y + 1)) ^ 0).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u * u ^ 0 / 1 + u * u ^ 2 / INR (fact 3) * -1 +
u * u ^ 4 / INR (fact 5) < u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  unfold Rdiv; rewrite Rinv_1, !Rmult_assoc, <- !Rmult_plus_distr_l.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u *
(u ^ 0 * 1 + u ^ 2 * (/ INR (fact 3) * -1) +
 u ^ 4 * / INR (fact 5)) < u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  apply tmp;[assumption | ].y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u ^ 0 * 1 + u ^ 2 * (/ INR (fact 3) * -1) +
u ^ 4 * / INR (fact 5) < 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  rewrite Rplus_assoc, Rmult_1_l; pattern 1 at 2; rewrite <- Rplus_0_r.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
1 +
(u ^ 2 * (/ INR (fact 3) * -1) +
 u ^ 4 * / INR (fact 5)) < 
1 + 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  apply Rplus_lt_compat_l.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u ^ 2 * (/ INR (fact 3) * -1) + u ^ 4 * / INR (fact 5) <
0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  rewrite <- Rmult_assoc.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u ^ 2 * / INR (fact 3) * -1 + u ^ 4 * / INR (fact 5) <
0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  match goal with |- (?a * (-1)) + _ < 0 =>
   rewrite <- (Rplus_opp_l a); change (-1) with (-(1)); rewrite Ropp_mult_distr_r_reverse, Rmult_1_r
  end.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
- (u ^ 2 * / INR (fact 3)) + u ^ 4 * / INR (fact 5) <
- (u ^ 2 * / INR (fact 3)) + u ^ 2 * / INR (fact 3)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  apply Rplus_lt_compat_l.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u ^ 4 * / INR (fact 5) < u ^ 2 * / INR (fact 3)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  assert (0 < u ^ 2) by (apply pow_lt; assumption).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
u ^ 4 * / INR (fact 5) < u ^ 2 * / INR (fact 3)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  replace (u ^ 4) with (u ^ 2 * u ^ 2) by ring.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
u ^ 2 * u ^ 2 * / INR (fact 5) <
u ^ 2 * / INR (fact 3)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
 rewrite Rmult_assoc; apply Rmult_lt_compat_l; auto.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
u ^ 2 * / INR (fact 5) < / INR (fact 3)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  apply Rlt_trans with (u ^ 2 * /INR (fact 3)).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
u ^ 2 * / INR (fact 5) < u ^ 2 * / INR (fact 3)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
u ^ 2 * / INR (fact 3) < / INR (fact 3)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
   apply Rmult_lt_compat_l; auto.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
/ INR (fact 5) < / INR (fact 3)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
u ^ 2 * / INR (fact 3) < / INR (fact 3)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
   apply Rinv_lt_contravar.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
0 < INR (fact 3) * INR (fact 5)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
INR (fact 3) < INR (fact 5)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
u ^ 2 * / INR (fact 3) < / INR (fact 3)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
    solve[apply Rmult_lt_0_compat; apply INR_fact_lt_0].y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
INR (fact 3) < INR (fact 5)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
u ^ 2 * / INR (fact 3) < / INR (fact 3)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
   rewrite !INR_IZR_INZ; apply IZR_lt; reflexivity.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
u ^ 2 * / INR (fact 3) < / INR (fact 3)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  rewrite Rmult_comm; apply tmp.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
0 < / INR (fact 3)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
u ^ 2 < 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
   solve[apply Rinv_0_lt_compat, INR_fact_lt_0].y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
u ^ 2 < 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  apply Rlt_trans with (2 := vlt2).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:0 < u ^ 2
u ^ 2 < u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  simpl; unfold u; apply tmp; auto; rewrite Rmult_1_r; assumption.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
 apply Rlt_trans with (Rabs y + 1);[lra | ].y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
Rabs y + 1 < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
 pattern (Rabs y + 1) at 1; rewrite <- (Rinv_involutive (Rabs y + 1));
  [ | apply Rgt_not_eq; lra].y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
/ / (Rabs y + 1) < / 2 * / sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
 rewrite <- Rinv_mult_distr.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
/ / (Rabs y + 1) < / (2 * sin u)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
sin u <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
   apply Rinv_lt_contravar.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
0 < 2 * sin u * / (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 * sin u < / (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
sin u <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
    apply Rmult_lt_0_compat.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
0 < 2 * sin u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
0 < / (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 * sin u < / (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
sin u <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
     apply Rmult_lt_0_compat;[lra | assumption].y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
0 < / (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 * sin u < / (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
sin u <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
    assumption.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 * sin u < / (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
sin u <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
   replace (/(Rabs y + 1)) with (2 * u).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 * sin u < 2 * u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 * u = / (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
sin u <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
    lra.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 * u = / (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
sin u <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
   unfold u; field; apply Rgt_not_eq; clear -Hgt; lra.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
2 <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
sin u <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
  solve[discrR].y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
H2:sin u < u
sin u <> 0
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
 apply Rgt_not_eq; assumption.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) < tan (PI / 2 - u)
unfold tan.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
/ 2 * / cos (PI / 2 - u) <
sin (PI / 2 - u) / cos (PI / 2 - u)
set (u' := PI / 2); unfold Rdiv; apply Rmult_lt_compat_r; unfold u'.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
0 < / cos (PI / 2 - u)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
/ 2 < sin (PI / 2 - u)
 apply Rinv_0_lt_compat.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
0 < cos (PI / 2 - u)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
/ 2 < sin (PI / 2 - u) 
 rewrite cos_shift; assumption.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
/ 2 < sin (PI / 2 - u)
assert (vlt3 : u < /4).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
u < / 4
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
/ 2 < sin (PI / 2 - u)
 replace (/4) with (/2 * /2) by field.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
u < / 2 * / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
/ 2 < sin (PI / 2 - u)
 unfold u; apply Rmult_lt_compat_l;[lra | ].y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
/ (Rabs y + 1) < / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
/ 2 < sin (PI / 2 - u)
 apply Rinv_lt_contravar.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
0 < 2 * (Rabs y + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
2 < Rabs y + 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
/ 2 < sin (PI / 2 - u)
  apply Rmult_lt_0_compat; lra.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
2 < Rabs y + 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
/ 2 < sin (PI / 2 - u)
 lra.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
/ 2 < sin (PI / 2 - u)
assert (1 < PI / 2 - u) by (assert (t := PI2_3_2); lra).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
/ 2 < sin (PI / 2 - u)
apply Rlt_trans with (sin 1).y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
/ 2 < sin 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
sin 1 < sin (PI / 2 - u)
 assert (t' : 1 <= 4) by lra.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
t':1 <= 4
/ 2 < sin 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
sin 1 < sin (PI / 2 - u)
 destruct (pre_sin_bound 1 0 (Rlt_le _ _ Rlt_0_1) t') as [t _].y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
t':1 <= 4
t:sin_approx 1 (2 * 0 + 1) <= sin 1
/ 2 < sin 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
sin 1 < sin (PI / 2 - u)
 apply Rlt_le_trans with (2 := t); clear t.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
t':1 <= 4
/ 2 < sin_approx 1 (2 * 0 + 1)
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
sin 1 < sin (PI / 2 - u)
 simpl plus; replace (sin_approx 1 1) with (5/6);[lra | ].y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
t':1 <= 4
5 / 6 = sin_approx 1 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
sin 1 < sin (PI / 2 - u)
 unfold sin_approx, sin_term; simpl; field.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
sin 1 < sin (PI / 2 - u)
apply sin_increasing_1.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
- (PI / 2) <= 1
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
1 <= PI / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
- (PI / 2) <= PI / 2 - u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
PI / 2 - u <= PI / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
1 < PI / 2 - u
    assert (t := PI2_1); lra.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
1 <= PI / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
- (PI / 2) <= PI / 2 - u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
PI / 2 - u <= PI / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
1 < PI / 2 - u
   apply Rlt_le, PI2_1.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
- (PI / 2) <= PI / 2 - u
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
PI / 2 - u <= PI / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
1 < PI / 2 - u
  assert (t := PI2_1); lra.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
PI / 2 - u <= PI / 2
y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
1 < PI / 2 - u
 lra.y:R
Hgt:Rabs y > 1
H:0 < / (Rabs y + 1)
u:=/ 2 * / (Rabs y + 1):R
H0:0 < u
vlt1:/ (Rabs y + 1) < 1
vlt2:u < 1
int:0 < PI / 2 - u < PI / 2
tmp:forall x y0 : R, 0 < x -> y0 < 1 -> x * y0 < x
H1:0 < sin u
u':=PI / 2:R
vlt3:u < / 4
H2:1 < PI / 2 - u
1 < PI / 2 - u
assumption.
Qed.

Lemma ub_opp : forall x, x < PI/2 -> -PI/2 < -x.forall x : R, x < PI / 2 -> - PI / 2 < - x
Proof.forall x : R, x < PI / 2 -> - PI / 2 < - x
intros x h; rewrite Ropp_div; apply Ropp_lt_contravar; assumption.
Qed.

Lemma pos_opp_lt : forall x, 0 < x -> -x < x.forall x : R, 0 < x -> - x < x
Proof.forall x : R, 0 < x -> - x < x intros; lra. Qed.

Lemma tech_opp_tan : forall x y, -tan x < y -> tan (-x) < y.forall x y : R, - tan x < y -> tan (- x) < y
Proof.forall x y : R, - tan x < y -> tan (- x) < y
intros; rewrite tan_neg; assumption.
Qed.

Definition pre_atan (y : R) : {x : R | -PI/2 < x < PI/2 /\ tan x = y}.y:R
{x : R | - PI / 2 < x < PI / 2 /\ tan x = y}
Proof.y:R
{x : R | - PI / 2 < x < PI / 2 /\ tan x = y}
destruct (frame_tan y) as [ub [[ub0 ubpi2] Ptan_ub]].y, ub:R
ub0:0 < ub
ubpi2:ub < PI / 2
Ptan_ub:Rabs y < tan ub
{x : R | - PI / 2 < x < PI / 2 /\ tan x = y}
set (pr := (conj (tech_opp_tan _ _ (proj2 (Rabs_def2 _ _ Ptan_ub)))
     (proj1 (Rabs_def2 _ _ Ptan_ub)))).y, ub:R
ub0:0 < ub
ubpi2:ub < PI / 2
Ptan_ub:Rabs y < tan ub
pr:=conj
  (tech_opp_tan ub y
     (proj2 (Rabs_def2 y (tan ub) Ptan_ub)))
  (proj1 (Rabs_def2 y (tan ub) Ptan_ub)):tan (- ub) < y < tan ub
{x : R | - PI / 2 < x < PI / 2 /\ tan x = y}
destruct (exists_atan_in_frame (-ub) ub y (pos_opp_lt _ ub0) (ub_opp _ ubpi2)
             ubpi2 pr) as [v [[vl vu] vq]].y, ub:R
ub0:0 < ub
ubpi2:ub < PI / 2
Ptan_ub:Rabs y < tan ub
pr:=conj
  (tech_opp_tan ub y
     (proj2 (Rabs_def2 y (tan ub) Ptan_ub)))
  (proj1 (Rabs_def2 y (tan ub) Ptan_ub)):tan (- ub) < y < tan ub
v:R
vl:- ub < v
vu:v < ub
vq:tan v = y
{x : R | - PI / 2 < x < PI / 2 /\ tan x = y}
exists v; clear pr.y, ub:R
ub0:0 < ub
ubpi2:ub < PI / 2
Ptan_ub:Rabs y < tan ub
v:R
vl:- ub < v
vu:v < ub
vq:tan v = y
- PI / 2 < v < PI / 2 /\ tan v = y
split;[rewrite Ropp_div; split; lra | assumption].
Qed.

Definition atan x := let (v, _) := pre_atan x in v.

Lemma atan_bound : forall x, -PI/2 < atan x < PI/2.forall x : R, - PI / 2 < atan x < PI / 2
Proof.forall x : R, - PI / 2 < atan x < PI / 2
intros x; unfold atan; destruct (pre_atan x) as [v [int _]]; exact int.
Qed.

Lemma atan_right_inv : forall x, tan (atan x) = x.forall x : R, tan (atan x) = x
Proof.forall x : R, tan (atan x) = x
intros x; unfold atan; destruct (pre_atan x) as [v [_ q]]; exact q.
Qed.

Lemma atan_opp : forall x, atan (- x) = - atan x.forall x : R, atan (- x) = - atan x
Proof.forall x : R, atan (- x) = - atan x
intros x; generalize (atan_bound (-x)); rewrite Ropp_div;intros [a b].x:R
a:- (PI / 2) < atan (- x)
b:atan (- x) < PI / 2
atan (- x) = - atan x
generalize (atan_bound x); rewrite Ropp_div; intros [c d].x:R
a:- (PI / 2) < atan (- x)
b:atan (- x) < PI / 2
c:- (PI / 2) < atan x
d:atan x < PI / 2
atan (- x) = - atan x
apply tan_is_inj; try rewrite Ropp_div; try split; try lra.x:R
a:- (PI / 2) < atan (- x)
b:atan (- x) < PI / 2
c:- (PI / 2) < atan x
d:atan x < PI / 2
tan (atan (- x)) = tan (- atan x)
rewrite tan_neg, !atan_right_inv; reflexivity.
Qed.

Lemma derivable_pt_atan : forall x, derivable_pt atan x.forall x : R, derivable_pt atan x
Proof.forall x : R, derivable_pt atan x
intros x.x:R
derivable_pt atan x
destruct (frame_tan x) as [ub [[ub0 ubpi] P]].x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
derivable_pt atan x
assert (lb_lt_ub : -ub < ub) by apply pos_opp_lt, ub0.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
derivable_pt atan x
assert (xint : tan(-ub) < x < tan ub).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
tan (- ub) < x < tan ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
derivable_pt atan x
 assert (xint' : x < tan ub /\ -(tan ub) < x) by apply Rabs_def2, P.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint':x < tan ub /\ - tan ub < x
tan (- ub) < x < tan ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
derivable_pt atan x
  rewrite tan_neg; tauto.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
derivable_pt atan x
assert (inv_p : forall x, tan(-ub) <= x -> x <= tan ub -> 
                     comp tan atan x = id x).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
derivable_pt atan x
 intros; apply atan_right_inv.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
derivable_pt atan x
assert (int_tan : forall y, tan (- ub) <= y -> y <= tan ub ->
                       -ub <= atan y <= ub).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
forall y : R,
tan (- ub) <= y -> y <= tan ub -> - ub <= atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
 clear -ub0 ubpi; intros y lo up; split.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
  destruct (Rle_lt_dec (-ub) (atan y)) as [h | abs]; auto.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
  assert (y < tan (-ub)).ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
y < tan (- ub)
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
H:y < tan (- ub)
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
   rewrite <- (atan_right_inv y); apply tan_increasing.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
- PI / 2 < atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
atan y < - ub
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
- ub < PI / 2
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
H:y < tan (- ub)
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
     destruct (atan_bound y); assumption.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
atan y < - ub
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
- ub < PI / 2
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
H:y < tan (- ub)
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
    assumption.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
- ub < PI / 2
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
H:y < tan (- ub)
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
   lra.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
H:y < tan (- ub)
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
  lra.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
 destruct (Rle_lt_dec (atan y) ub) as [h | abs]; auto.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
  assert (tan ub < y).ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
tan ub < y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
H:tan ub < y
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
   rewrite <- (atan_right_inv y); apply tan_increasing.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
- PI / 2 < ub
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
ub < atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
atan y < PI / 2
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
H:tan ub < y
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
    rewrite Ropp_div; lra.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
ub < atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
atan y < PI / 2
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
H:tan ub < y
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
   assumption.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
atan y < PI / 2
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
H:tan ub < y
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
  destruct (atan_bound y); assumption.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
H:tan ub < y
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
 lra.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derivable_pt atan x
assert (incr : forall x y, -ub <= x -> x < y -> y <= ub -> tan x < tan y).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
forall x0 y : R,
- ub <= x0 -> x0 < y -> y <= ub -> tan x0 < tan y
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
derivable_pt atan x
 intros y z l yz u; apply tan_increasing.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y0 : R,
tan (- ub) <= y0 ->
y0 <= tan ub -> - ub <= atan y0 <= ub
y, z:R
l:- ub <= y
yz:y < z
u:z <= ub
- PI / 2 < y
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y0 : R,
tan (- ub) <= y0 ->
y0 <= tan ub -> - ub <= atan y0 <= ub
y, z:R
l:- ub <= y
yz:y < z
u:z <= ub
y < z
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y0 : R,
tan (- ub) <= y0 ->
y0 <= tan ub -> - ub <= atan y0 <= ub
y, z:R
l:- ub <= y
yz:y < z
u:z <= ub
z < PI / 2
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
derivable_pt atan x
   rewrite Ropp_div; lra.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y0 : R,
tan (- ub) <= y0 ->
y0 <= tan ub -> - ub <= atan y0 <= ub
y, z:R
l:- ub <= y
yz:y < z
u:z <= ub
y < z
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y0 : R,
tan (- ub) <= y0 ->
y0 <= tan ub -> - ub <= atan y0 <= ub
y, z:R
l:- ub <= y
yz:y < z
u:z <= ub
z < PI / 2
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
derivable_pt atan x
  assumption.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y0 : R,
tan (- ub) <= y0 ->
y0 <= tan ub -> - ub <= atan y0 <= ub
y, z:R
l:- ub <= y
yz:y < z
u:z <= ub
z < PI / 2
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
derivable_pt atan x
 lra.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
derivable_pt atan x
assert (der : forall a, -ub <= a <= ub -> derivable_pt tan a).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
forall a : R, - ub <= a <= ub -> derivable_pt tan a
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
derivable_pt atan x
 intros a [la ua]; apply derivable_pt_tan.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
a:R
la:- ub <= a
ua:a <= ub
- PI / 2 < a < PI / 2
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
derivable_pt atan x
 rewrite Ropp_div; split; lra.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
derivable_pt atan x
assert (df_neq : derive_pt tan (atan x)
     (derivable_pt_recip_interv_prelim1 tan atan 
        (- ub) ub x lb_lt_ub xint inv_p int_tan incr der) <> 0).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan 
     (- ub) ub x lb_lt_ub xint inv_p int_tan incr der) <>
0
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
derivable_pt atan x
 rewrite <- (pr_nu tan (atan x)
              (derivable_pt_tan (atan x) (atan_bound x))).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
derive_pt tan (atan x)
  (derivable_pt_tan (atan x) (atan_bound x)) <> 0
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
derivable_pt atan x
 rewrite derive_pt_tan.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
1 + tan (atan x) ^ 2 <> 0
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
derivable_pt atan x
 solve[apply Rgt_not_eq, plus_Rsqr_gt_0].x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
derivable_pt atan x
apply (derivable_pt_recip_interv tan atan (-ub) ub x
      lb_lt_ub xint inv_p int_tan incr der).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
P:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan 
     (- ub) ub x lb_lt_ub xint inv_p int_tan incr der) <>
0
exact df_neq.
Qed.

Lemma atan_increasing : forall x y, x < y -> atan x < atan y.forall x y : R, x < y -> atan x < atan y
Proof.forall x y : R, x < y -> atan x < atan y
intros x y d.x, y:R
d:x < y
atan x < atan y
assert (t1 := atan_bound x).x, y:R
d:x < y
t1:- PI / 2 < atan x < PI / 2
atan x < atan y
assert (t2 := atan_bound y).x, y:R
d:x < y
t1:- PI / 2 < atan x < PI / 2
t2:- PI / 2 < atan y < PI / 2
atan x < atan y
destruct (Rlt_le_dec (atan x) (atan y)) as [lt | bad].x, y:R
d:x < y
t1:- PI / 2 < atan x < PI / 2
t2:- PI / 2 < atan y < PI / 2
lt:atan x < atan y
atan x < atan y
x, y:R
d:x < y
t1:- PI / 2 < atan x < PI / 2
t2:- PI / 2 < atan y < PI / 2
bad:atan y <= atan x
atan x < atan y
 assumption.x, y:R
d:x < y
t1:- PI / 2 < atan x < PI / 2
t2:- PI / 2 < atan y < PI / 2
bad:atan y <= atan x
atan x < atan y
apply Rlt_not_le in d.x, y:R
d:~ y <= x
t1:- PI / 2 < atan x < PI / 2
t2:- PI / 2 < atan y < PI / 2
bad:atan y <= atan x
atan x < atan y
case d.x, y:R
d:~ y <= x
t1:- PI / 2 < atan x < PI / 2
t2:- PI / 2 < atan y < PI / 2
bad:atan y <= atan x
y <= x
rewrite <- (atan_right_inv y), <- (atan_right_inv x).x, y:R
d:~ y <= x
t1:- PI / 2 < atan x < PI / 2
t2:- PI / 2 < atan y < PI / 2
bad:atan y <= atan x
tan (atan y) <= tan (atan x)
destruct bad as [ylt | yx].x, y:R
d:~ y <= x
t1:- PI / 2 < atan x < PI / 2
t2:- PI / 2 < atan y < PI / 2
ylt:atan y < atan x
tan (atan y) <= tan (atan x)
x, y:R
d:~ y <= x
t1:- PI / 2 < atan x < PI / 2
t2:- PI / 2 < atan y < PI / 2
yx:atan y = atan x
tan (atan y) <= tan (atan x)
 apply Rlt_le, tan_increasing; try tauto.x, y:R
d:~ y <= x
t1:- PI / 2 < atan x < PI / 2
t2:- PI / 2 < atan y < PI / 2
yx:atan y = atan x
tan (atan y) <= tan (atan x)
solve[rewrite yx; apply Rle_refl].
Qed.

Lemma atan_0 : atan 0 = 0.atan 0 = 0
Proof.atan 0 = 0
apply tan_is_inj; try (apply atan_bound).- PI / 2 < 0 < PI / 2
tan (atan 0) = tan 0
 assert (t := PI_RGT_0); rewrite Ropp_div; split; lra.tan (atan 0) = tan 0
rewrite atan_right_inv, tan_0.0 = 0
reflexivity.
Qed.

Lemma atan_1 : atan 1 = PI/4.atan 1 = PI / 4
Proof.atan 1 = PI / 4
assert (ut := PI_RGT_0).ut:PI > 0
atan 1 = PI / 4
assert (-PI/2 < PI/4 < PI/2) by (rewrite Ropp_div; split; lra).ut:PI > 0
H:- PI / 2 < PI / 4 < PI / 2
atan 1 = PI / 4
assert (t := atan_bound 1).ut:PI > 0
H:- PI / 2 < PI / 4 < PI / 2
t:- PI / 2 < atan 1 < PI / 2
atan 1 = PI / 4
apply tan_is_inj; auto.ut:PI > 0
H:- PI / 2 < PI / 4 < PI / 2
t:- PI / 2 < atan 1 < PI / 2
tan (atan 1) = tan (PI / 4)
rewrite tan_PI4, atan_right_inv; reflexivity.
Qed.

atan's derivative value is the function 1 / (1+x²)

Lemma derive_pt_atan : forall x,
      derive_pt atan x (derivable_pt_atan x) =
         1 / (1 + x²).forall x : R,
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
Proof.forall x : R,
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
intros x.x:R
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
destruct (frame_tan x) as [ub [[ub0 ubpi] Pub]].x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
assert (lb_lt_ub : -ub < ub) by apply pos_opp_lt, ub0.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
assert (xint : tan(-ub) < x < tan ub).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
tan (- ub) < x < tan ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
 assert (xint' : x < tan ub /\ -(tan ub) < x) by apply Rabs_def2, Pub.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint':x < tan ub /\ - tan ub < x
tan (- ub) < x < tan ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
  rewrite tan_neg; tauto.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
assert (inv_p : forall x, tan(-ub) <= x -> x <= tan ub -> 
                     comp tan atan x = id x).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
 intros; apply atan_right_inv.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
assert (int_tan : forall y, tan (- ub) <= y -> y <= tan ub ->
                       -ub <= atan y <= ub).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
forall y : R,
tan (- ub) <= y -> y <= tan ub -> - ub <= atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
 clear -ub0 ubpi; intros y lo up; split.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
  destruct (Rle_lt_dec (-ub) (atan y)) as [h | abs]; auto.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
  assert (y < tan (-ub)).ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
y < tan (- ub)
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
H:y < tan (- ub)
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
   rewrite <- (atan_right_inv y); apply tan_increasing.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
- PI / 2 < atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
atan y < - ub
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
- ub < PI / 2
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
H:y < tan (- ub)
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
     destruct (atan_bound y); assumption.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
atan y < - ub
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
- ub < PI / 2
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
H:y < tan (- ub)
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
    assumption.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
- ub < PI / 2
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
H:y < tan (- ub)
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
   lra.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:atan y < - ub
H:y < tan (- ub)
- ub <= atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
  lra.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
 destruct (Rle_lt_dec (atan y) ub) as [h | abs]; auto.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
  assert (tan ub < y).ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
tan ub < y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
H:tan ub < y
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
   rewrite <- (atan_right_inv y); apply tan_increasing.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
- PI / 2 < ub
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
ub < atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
atan y < PI / 2
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
H:tan ub < y
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
    rewrite Ropp_div; lra.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
ub < atan y
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
atan y < PI / 2
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
H:tan ub < y
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
   assumption.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
atan y < PI / 2
ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
H:tan ub < y
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
  destruct (atan_bound y); assumption.ub:R
ub0:0 < ub
ubpi:ub < PI / 2
y:R
lo:tan (- ub) <= y
up:y <= tan ub
abs:ub < atan y
H:tan ub < y
atan y <= ub
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
 lra.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
assert (incr : forall x y, -ub <= x -> x < y -> y <= ub -> tan x < tan y).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
forall x0 y : R,
- ub <= x0 -> x0 < y -> y <= ub -> tan x0 < tan y
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
 intros y z l yz u; apply tan_increasing.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y0 : R,
tan (- ub) <= y0 ->
y0 <= tan ub -> - ub <= atan y0 <= ub
y, z:R
l:- ub <= y
yz:y < z
u:z <= ub
- PI / 2 < y
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y0 : R,
tan (- ub) <= y0 ->
y0 <= tan ub -> - ub <= atan y0 <= ub
y, z:R
l:- ub <= y
yz:y < z
u:z <= ub
y < z
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y0 : R,
tan (- ub) <= y0 ->
y0 <= tan ub -> - ub <= atan y0 <= ub
y, z:R
l:- ub <= y
yz:y < z
u:z <= ub
z < PI / 2
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
   rewrite Ropp_div; lra.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y0 : R,
tan (- ub) <= y0 ->
y0 <= tan ub -> - ub <= atan y0 <= ub
y, z:R
l:- ub <= y
yz:y < z
u:z <= ub
y < z
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y0 : R,
tan (- ub) <= y0 ->
y0 <= tan ub -> - ub <= atan y0 <= ub
y, z:R
l:- ub <= y
yz:y < z
u:z <= ub
z < PI / 2
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
  assumption.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y0 : R,
tan (- ub) <= y0 ->
y0 <= tan ub -> - ub <= atan y0 <= ub
y, z:R
l:- ub <= y
yz:y < z
u:z <= ub
z < PI / 2
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
 lra.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
assert (der : forall a, -ub <= a <= ub -> derivable_pt tan a).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
forall a : R, - ub <= a <= ub -> derivable_pt tan a
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
 intros a [la ua]; apply derivable_pt_tan.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
a:R
la:- ub <= a
ua:a <= ub
- PI / 2 < a < PI / 2
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
 rewrite Ropp_div; split; lra.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
assert (df_neq : derive_pt tan (atan x)
     (derivable_pt_recip_interv_prelim1 tan atan 
        (- ub) ub x lb_lt_ub xint inv_p int_tan incr der) <> 0).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan 
     (- ub) ub x lb_lt_ub xint inv_p int_tan incr der) <>
0
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
 rewrite <- (pr_nu tan (atan x)
              (derivable_pt_tan (atan x) (atan_bound x))).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
derive_pt tan (atan x)
  (derivable_pt_tan (atan x) (atan_bound x)) <> 0
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
 rewrite derive_pt_tan.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
1 + tan (atan x) ^ 2 <> 0
x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
 solve[apply Rgt_not_eq, plus_Rsqr_gt_0].x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
assert (t := derive_pt_recip_interv tan atan (-ub) ub x lb_lt_ub
        xint incr int_tan der inv_p df_neq).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
t:derive_pt atan x
  (derivable_pt_recip_interv tan atan 
     (- ub) ub x lb_lt_ub xint inv_p int_tan incr
     der df_neq) =
1 /
derive_pt tan (atan x)
  (der (atan x)
     (derive_pt_recip_interv_prelim1_1 tan atan
        (- ub) ub x lb_lt_ub xint incr int_tan
        inv_p))
derive_pt atan x (derivable_pt_atan x) = 1 / (1 + x²)
rewrite <- (pr_nu atan x (derivable_pt_recip_interv tan atan (- ub) ub
      x lb_lt_ub xint inv_p int_tan incr der df_neq)).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
t:derive_pt atan x
  (derivable_pt_recip_interv tan atan 
     (- ub) ub x lb_lt_ub xint inv_p int_tan incr
     der df_neq) =
1 /
derive_pt tan (atan x)
  (der (atan x)
     (derive_pt_recip_interv_prelim1_1 tan atan
        (- ub) ub x lb_lt_ub xint incr int_tan
        inv_p))
derive_pt atan x
  (derivable_pt_recip_interv tan atan 
     (- ub) ub x lb_lt_ub xint inv_p int_tan incr der
     df_neq) = 1 / (1 + x²)
rewrite t.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
t:derive_pt atan x
  (derivable_pt_recip_interv tan atan 
     (- ub) ub x lb_lt_ub xint inv_p int_tan incr
     der df_neq) =
1 /
derive_pt tan (atan x)
  (der (atan x)
     (derive_pt_recip_interv_prelim1_1 tan atan
        (- ub) ub x lb_lt_ub xint incr int_tan
        inv_p))
1 /
derive_pt tan (atan x)
  (der (atan x)
     (derive_pt_recip_interv_prelim1_1 tan atan 
        (- ub) ub x lb_lt_ub xint incr int_tan inv_p)) =
1 / (1 + x²)
assert (t' := atan_bound x).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
t:derive_pt atan x
  (derivable_pt_recip_interv tan atan 
     (- ub) ub x lb_lt_ub xint inv_p int_tan incr
     der df_neq) =
1 /
derive_pt tan (atan x)
  (der (atan x)
     (derive_pt_recip_interv_prelim1_1 tan atan
        (- ub) ub x lb_lt_ub xint incr int_tan
        inv_p))
t':- PI / 2 < atan x < PI / 2
1 /
derive_pt tan (atan x)
  (der (atan x)
     (derive_pt_recip_interv_prelim1_1 tan atan 
        (- ub) ub x lb_lt_ub xint incr int_tan inv_p)) =
1 / (1 + x²)
rewrite <- (pr_nu tan (atan x) (derivable_pt_tan _ t')).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
t:derive_pt atan x
  (derivable_pt_recip_interv tan atan 
     (- ub) ub x lb_lt_ub xint inv_p int_tan incr
     der df_neq) =
1 /
derive_pt tan (atan x)
  (der (atan x)
     (derive_pt_recip_interv_prelim1_1 tan atan
        (- ub) ub x lb_lt_ub xint incr int_tan
        inv_p))
t':- PI / 2 < atan x < PI / 2
1 /
derive_pt tan (atan x) (derivable_pt_tan (atan x) t') =
1 / (1 + x²) 
rewrite derive_pt_tan, atan_right_inv.x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
t:derive_pt atan x
  (derivable_pt_recip_interv tan atan 
     (- ub) ub x lb_lt_ub xint inv_p int_tan incr
     der df_neq) =
1 /
derive_pt tan (atan x)
  (der (atan x)
     (derive_pt_recip_interv_prelim1_1 tan atan
        (- ub) ub x lb_lt_ub xint incr int_tan
        inv_p))
t':- PI / 2 < atan x < PI / 2
1 / (1 + x ^ 2) = 1 / (1 + x²)
replace (Rsqr x) with (x ^ 2) by (unfold Rsqr; ring).x, ub:R
ub0:0 < ub
ubpi:ub < PI / 2
Pub:Rabs x < tan ub
lb_lt_ub:- ub < ub
xint:tan (- ub) < x < tan ub
inv_p:forall x0 : R,
tan (- ub) <= x0 ->
x0 <= tan ub -> comp tan atan x0 = id x0
int_tan:forall y : R,
tan (- ub) <= y ->
y <= tan ub -> - ub <= atan y <= ub
incr:forall x0 y : R,
- ub <= x0 ->
x0 < y -> y <= ub -> tan x0 < tan y
der:forall a : R,
- ub <= a <= ub -> derivable_pt tan a
df_neq:derive_pt tan (atan x)
  (derivable_pt_recip_interv_prelim1 tan atan
     (- ub) ub x lb_lt_ub xint inv_p int_tan
     incr der) <> 0
t:derive_pt atan x
  (derivable_pt_recip_interv tan atan 
     (- ub) ub x lb_lt_ub xint inv_p int_tan incr
     der df_neq) =
1 /
derive_pt tan (atan x)
  (der (atan x)
     (derive_pt_recip_interv_prelim1_1 tan atan
        (- ub) ub x lb_lt_ub xint incr int_tan
        inv_p))
t':- PI / 2 < atan x < PI / 2
1 / (1 + x ^ 2) = 1 / (1 + x ^ 2)
reflexivity.
Qed.

Lemma derivable_pt_lim_atan :
  forall x, derivable_pt_lim atan x (/(1 + x^2)).forall x : R, derivable_pt_lim atan x (/ (1 + x ^ 2))
Proof.forall x : R, derivable_pt_lim atan x (/ (1 + x ^ 2))
intros x.x:R
derivable_pt_lim atan x (/ (1 + x ^ 2))
apply derive_pt_eq_1 with (derivable_pt_atan x).x:R
derive_pt atan x (derivable_pt_atan x) = / (1 + x ^ 2)
replace (x ^ 2) with (x * x) by ring.x:R
derive_pt atan x (derivable_pt_atan x) = / (1 + x * x)
rewrite <- (Rmult_1_l (Rinv _)).x:R
derive_pt atan x (derivable_pt_atan x) =
1 * / (1 + x * x)
apply derive_pt_atan.
Qed.

Definition of the arctangent function as the sum of the arctan power series

(* Proof taken from Guillaume Melquiond's interval package for Coq *)

Definition Ratan_seq x :=  fun n => (x ^ (2 * n + 1) / INR (2 * n + 1))%R.

Lemma Ratan_seq_decreasing : forall x, (0 <= x <= 1)%R -> Un_decreasing (Ratan_seq x).forall x : R,
0 <= x <= 1 -> Un_decreasing (Ratan_seq x)
Proof.forall x : R,
0 <= x <= 1 -> Un_decreasing (Ratan_seq x)
intros x Hx n.x:R
Hx:0 <= x <= 1
n:nat
Ratan_seq x (S n) <= Ratan_seq x n
 unfold Ratan_seq, Rdiv.x:R
Hx:0 <= x <= 1
n:nat
x ^ (2 * S n + 1) * / INR (2 * S n + 1) <=
x ^ (2 * n + 1) * / INR (2 * n + 1)
 apply Rmult_le_compat.x:R
Hx:0 <= x <= 1
n:nat
0 <= x ^ (2 * S n + 1)
x:R
Hx:0 <= x <= 1
n:nat
0 <= / INR (2 * S n + 1)
x:R
Hx:0 <= x <= 1
n:nat
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1) apply pow_le.x:R
Hx:0 <= x <= 1
n:nat
0 <= x
x:R
Hx:0 <= x <= 1
n:nat
0 <= / INR (2 * S n + 1)
x:R
Hx:0 <= x <= 1
n:nat
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 exact (proj1 Hx).x:R
Hx:0 <= x <= 1
n:nat
0 <= / INR (2 * S n + 1)
x:R
Hx:0 <= x <= 1
n:nat
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 apply Rlt_le.x:R
Hx:0 <= x <= 1
n:nat
0 < / INR (2 * S n + 1)
x:R
Hx:0 <= x <= 1
n:nat
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 apply Rinv_0_lt_compat.x:R
Hx:0 <= x <= 1
n:nat
0 < INR (2 * S n + 1)
x:R
Hx:0 <= x <= 1
n:nat
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 apply lt_INR_0.x:R
Hx:0 <= x <= 1
n:nat
(0 < 2 * S n + 1)%nat
x:R
Hx:0 <= x <= 1
n:nat
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 omega.x:R
Hx:0 <= x <= 1
n:nat
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 destruct (proj1 Hx) as [Hx1|Hx1].x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 destruct (proj2 Hx) as [Hx2|Hx2].x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 (* . 0 < x < 1 *)
 rewrite <- (Rinv_involutive x).x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
(/ / x) ^ (2 * S n + 1) <= (/ / x) ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x <> 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 assert (/ x <> 0)%R by auto with real.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
(/ / x) ^ (2 * S n + 1) <= (/ / x) ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x <> 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 repeat rewrite <- Rinv_pow with (1 := H).x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
/ (/ x) ^ (2 * S n + 1) <= / (/ x) ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x <> 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 apply Rlt_le.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
/ (/ x) ^ (2 * S n + 1) < / (/ x) ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x <> 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 apply Rinv_lt_contravar.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
0 < (/ x) ^ (2 * n + 1) * (/ x) ^ (2 * S n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
(/ x) ^ (2 * n + 1) < (/ x) ^ (2 * S n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x <> 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 apply Rmult_lt_0_compat ; apply pow_lt ; auto with real.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
(/ x) ^ (2 * n + 1) < (/ x) ^ (2 * S n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x <> 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 apply Rlt_pow.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
1 < / x
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
(2 * n + 1 < 2 * S n + 1)%nat
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x <> 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 rewrite <- Rinv_1.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
/ 1 < / x
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
(2 * n + 1 < 2 * S n + 1)%nat
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x <> 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 apply Rinv_lt_contravar.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
0 < x * 1
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
x < 1
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
(2 * n + 1 < 2 * S n + 1)%nat
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x <> 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 rewrite Rmult_1_r.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
0 < x
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
x < 1
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
(2 * n + 1 < 2 * S n + 1)%nat
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x <> 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 exact Hx1.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
x < 1
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
(2 * n + 1 < 2 * S n + 1)%nat
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x <> 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 exact Hx2.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
H:/ x <> 0
(2 * n + 1 < 2 * S n + 1)%nat
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x <> 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 omega.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x <> 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 apply Rgt_not_eq.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x < 1
x > 0
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 exact Hx1.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 (* . x = 1 *)
 rewrite Hx2.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
1 ^ (2 * S n + 1) <= 1 ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 do 2 rewrite pow1.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 < x
Hx2:x = 1
1 <= 1
x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 apply Rle_refl.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
x ^ (2 * S n + 1) <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 (* . x = 0 *)
 rewrite <- Hx1.x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
0 ^ (2 * S n + 1) <= 0 ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 do 2 (rewrite pow_i ; [ idtac | omega ]).x:R
Hx:0 <= x <= 1
n:nat
Hx1:0 = x
0 <= 0
x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 apply Rle_refl.x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) <= / INR (2 * n + 1)
 apply Rlt_le.x:R
Hx:0 <= x <= 1
n:nat
/ INR (2 * S n + 1) < / INR (2 * n + 1)
 apply Rinv_lt_contravar.x:R
Hx:0 <= x <= 1
n:nat
0 < INR (2 * n + 1) * INR (2 * S n + 1)
x:R
Hx:0 <= x <= 1
n:nat
INR (2 * n + 1) < INR (2 * S n + 1)
 apply Rmult_lt_0_compat ; apply lt_INR_0 ; omega.x:R
Hx:0 <= x <= 1
n:nat
INR (2 * n + 1) < INR (2 * S n + 1)
 apply lt_INR.x:R
Hx:0 <= x <= 1
n:nat
(2 * n + 1 < 2 * S n + 1)%nat
 omega.
Qed.

Lemma Ratan_seq_converging : forall x, (0 <= x <= 1)%R -> Un_cv (Ratan_seq x) 0.forall x : R, 0 <= x <= 1 -> Un_cv (Ratan_seq x) 0
Proof.forall x : R, 0 <= x <= 1 -> Un_cv (Ratan_seq x) 0
intros x Hx eps Heps.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Ratan_seq x n) 0 < eps
 destruct (archimed (/ eps)) as (HN,_).x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
HN:IZR (up (/ eps)) > / eps
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Ratan_seq x n) 0 < eps
 assert (0 < up (/ eps))%Z.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
HN:IZR (up (/ eps)) > / eps
(0 < up (/ eps))%Z
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
HN:IZR (up (/ eps)) > / eps
H:(0 < up (/ eps))%Z
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Ratan_seq x n) 0 < eps
  apply lt_IZR.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
HN:IZR (up (/ eps)) > / eps
0 < IZR (up (/ eps))
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
HN:IZR (up (/ eps)) > / eps
H:(0 < up (/ eps))%Z
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Ratan_seq x n) 0 < eps
  apply Rlt_trans with (2 := HN).x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
HN:IZR (up (/ eps)) > / eps
0 < / eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
HN:IZR (up (/ eps)) > / eps
H:(0 < up (/ eps))%Z
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Ratan_seq x n) 0 < eps
  apply Rinv_0_lt_compat.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
HN:IZR (up (/ eps)) > / eps
0 < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
HN:IZR (up (/ eps)) > / eps
H:(0 < up (/ eps))%Z
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Ratan_seq x n) 0 < eps
  exact Heps.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
HN:IZR (up (/ eps)) > / eps
H:(0 < up (/ eps))%Z
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Ratan_seq x n) 0 < eps
 case_eq (up (/ eps)) ;
  intros ; rewrite H0 in H ; try discriminate H.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
HN:IZR (up (/ eps)) > / eps
p:positive
H:(0 < Z.pos p)%Z
H0:up (/ eps) = Z.pos p
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Ratan_seq x n) 0 < eps
   rewrite H0 in HN.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
H:(0 < Z.pos p)%Z
H0:up (/ eps) = Z.pos p
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Ratan_seq x n) 0 < eps
   simpl in HN.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
H:(0 < Z.pos p)%Z
H0:up (/ eps) = Z.pos p
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Ratan_seq x n) 0 < eps
   pose (N := Pos.to_nat p).x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
H:(0 < Z.pos p)%Z
H0:up (/ eps) = Z.pos p
N:=Pos.to_nat p:nat
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (Ratan_seq x n) 0 < eps
   fold N in HN.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
H:(0 < Z.pos p)%Z
H0:up (/ eps) = Z.pos p
N:=Pos.to_nat p:nat
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (Ratan_seq x n) 0 < eps
   clear H H0.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> R_dist (Ratan_seq x n) 0 < eps
   exists N.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
forall n : nat,
(n >= N)%nat -> R_dist (Ratan_seq x n) 0 < eps
  intros n Hn.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
R_dist (Ratan_seq x n) 0 < eps
   unfold R_dist.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
Rabs (Ratan_seq x n - 0) < eps
   rewrite Rminus_0_r.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
Rabs (Ratan_seq x n) < eps
   unfold Ratan_seq.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
Rabs (x ^ (2 * n + 1) / INR (2 * n + 1)) < eps
   rewrite Rabs_right.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply Rle_lt_trans with (1 ^ (2 * n + 1) / INR (2 * n + 1))%R.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) <=
1 ^ (2 * n + 1) / INR (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
1 ^ (2 * n + 1) / INR (2 * n + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   unfold Rdiv.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) * / INR (2 * n + 1) <=
1 ^ (2 * n + 1) * / INR (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
1 ^ (2 * n + 1) / INR (2 * n + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply Rmult_le_compat_r.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 <= / INR (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) <= 1 ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
1 ^ (2 * n + 1) / INR (2 * n + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply Rlt_le.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 < / INR (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) <= 1 ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
1 ^ (2 * n + 1) / INR (2 * n + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply Rinv_0_lt_compat.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 < INR (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) <= 1 ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
1 ^ (2 * n + 1) / INR (2 * n + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply lt_INR_0.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
(0 < 2 * n + 1)%nat
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) <= 1 ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
1 ^ (2 * n + 1) / INR (2 * n + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   omega.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) <= 1 ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
1 ^ (2 * n + 1) / INR (2 * n + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply pow_incr.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 <= x <= 1
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
1 ^ (2 * n + 1) / INR (2 * n + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   exact Hx.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
1 ^ (2 * n + 1) / INR (2 * n + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   rewrite pow1.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
1 / INR (2 * n + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply Rle_lt_trans with (/ INR (2 * N + 1))%R.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
1 / INR (2 * n + 1) <= / INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ INR (2 * N + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   unfold Rdiv.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
1 * / INR (2 * n + 1) <= / INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ INR (2 * N + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   rewrite Rmult_1_l.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ INR (2 * n + 1) <= / INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ INR (2 * N + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply Rinv_le_contravar.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
INR (2 * N + 1) <= INR (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ INR (2 * N + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply lt_INR_0.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
(0 < 2 * N + 1)%nat
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
INR (2 * N + 1) <= INR (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ INR (2 * N + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   omega.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
INR (2 * N + 1) <= INR (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ INR (2 * N + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply le_INR.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
(2 * N + 1 <= 2 * n + 1)%nat
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ INR (2 * N + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   omega.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ INR (2 * N + 1) < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   rewrite <- (Rinv_involutive eps).x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ INR (2 * N + 1) < / / eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply Rinv_lt_contravar.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 < / eps * INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ eps < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply Rmult_lt_0_compat.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 < / eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ eps < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   auto with real.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ eps < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply lt_INR_0.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
(0 < 2 * N + 1)%nat
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ eps < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   omega.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ eps < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply Rlt_trans with (INR N).x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
/ eps < INR N
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
INR N < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   destruct (archimed (/ eps)) as (H,_).x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
H:IZR (up (/ eps)) > / eps
/ eps < INR N
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
INR N < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   assert (0 < up (/ eps))%Z.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
H:IZR (up (/ eps)) > / eps
(0 < up (/ eps))%Z
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
H:IZR (up (/ eps)) > / eps
H0:(0 < up (/ eps))%Z
/ eps < INR N
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
INR N < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply lt_IZR.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
H:IZR (up (/ eps)) > / eps
0 < IZR (up (/ eps))
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
H:IZR (up (/ eps)) > / eps
H0:(0 < up (/ eps))%Z
/ eps < INR N
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
INR N < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply Rlt_trans with (2 := H).x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
H:IZR (up (/ eps)) > / eps
0 < / eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
H:IZR (up (/ eps)) > / eps
H0:(0 < up (/ eps))%Z
/ eps < INR N
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
INR N < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply Rinv_0_lt_compat.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
H:IZR (up (/ eps)) > / eps
0 < eps
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
H:IZR (up (/ eps)) > / eps
H0:(0 < up (/ eps))%Z
/ eps < INR N
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
INR N < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   exact Heps.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
H:IZR (up (/ eps)) > / eps
H0:(0 < up (/ eps))%Z
/ eps < INR N
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
INR N < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   unfold N.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
H:IZR (up (/ eps)) > / eps
H0:(0 < up (/ eps))%Z
/ eps < INR (Pos.to_nat p)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
INR N < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   rewrite INR_IZR_INZ, positive_nat_Z.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
H:IZR (up (/ eps)) > / eps
H0:(0 < up (/ eps))%Z
/ eps < IZR (Z.pos p)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
INR N < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   exact HN.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
INR N < INR (2 * N + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply lt_INR.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
(N < 2 * N + 1)%nat
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   omega.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps <> 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply Rgt_not_eq.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
eps > 0
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   exact Heps.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
x ^ (2 * n + 1) / INR (2 * n + 1) >= 0
   apply Rle_ge.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 <= x ^ (2 * n + 1) / INR (2 * n + 1)
   unfold Rdiv.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 <= x ^ (2 * n + 1) * / INR (2 * n + 1)
   apply Rmult_le_pos.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 <= x ^ (2 * n + 1)
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 <= / INR (2 * n + 1)
   apply pow_le.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 <= x
x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 <= / INR (2 * n + 1)
   exact (proj1 Hx).x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 <= / INR (2 * n + 1)
   apply Rlt_le.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 < / INR (2 * n + 1)
   apply Rinv_0_lt_compat.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
0 < INR (2 * n + 1)
   apply lt_INR_0.x:R
Hx:0 <= x <= 1
eps:R
Heps:eps > 0
p:positive
HN:IZR (Z.pos p) > / eps
N:=Pos.to_nat p:nat
n:nat
Hn:(n >= N)%nat
(0 < 2 * n + 1)%nat
   omega.
Qed.

Definition ps_atan_exists_01 (x : R) (Hx:0 <= x <= 1) :
   {l : R | Un_cv (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}.x:R
Hx:0 <= x <= 1
{l : R |
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}
Proof.x:R
Hx:0 <= x <= 1
{l : R |
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}
exact (alternated_series (Ratan_seq x)
  (Ratan_seq_decreasing _ Hx) (Ratan_seq_converging _ Hx)).
Defined.

Lemma Ratan_seq_opp : forall x n, Ratan_seq (-x) n = -Ratan_seq x n.forall (x : R) (n : nat),
Ratan_seq (- x) n = - Ratan_seq x n
Proof.forall (x : R) (n : nat),
Ratan_seq (- x) n = - Ratan_seq x n
intros x n; unfold Ratan_seq.x:R
n:nat
(- x) ^ (2 * n + 1) / INR (2 * n + 1) =
- (x ^ (2 * n + 1) / INR (2 * n + 1))
rewrite !pow_add, !pow_mult, !pow_1.x:R
n:nat
((- x) ^ 2) ^ n * - x / INR (2 * n + 1) =
- ((x ^ 2) ^ n * x / INR (2 * n + 1))
unfold Rdiv; replace ((-x) ^ 2) with (x ^ 2) by ring; ring.
Qed.

Lemma sum_Ratan_seq_opp : 
  forall x n, sum_f_R0 (tg_alt (Ratan_seq (- x))) n =
     - sum_f_R0 (tg_alt (Ratan_seq x)) n.forall (x : R) (n : nat),
sum_f_R0 (tg_alt (Ratan_seq (- x))) n =
- sum_f_R0 (tg_alt (Ratan_seq x)) n
Proof.forall (x : R) (n : nat),
sum_f_R0 (tg_alt (Ratan_seq (- x))) n =
- sum_f_R0 (tg_alt (Ratan_seq x)) n
intros x n; replace (-sum_f_R0 (tg_alt (Ratan_seq x)) n) with 
  (-1 * sum_f_R0 (tg_alt (Ratan_seq x)) n) by ring.x:R
n:nat
sum_f_R0 (tg_alt (Ratan_seq (- x))) n =
-1 * sum_f_R0 (tg_alt (Ratan_seq x)) n
rewrite scal_sum; apply sum_eq; intros i _; unfold tg_alt.x:R
n, i:nat
(-1) ^ i * Ratan_seq (- x) i =
(-1) ^ i * Ratan_seq x i * -1
rewrite Ratan_seq_opp; ring.
Qed.

Definition ps_atan_exists_1 (x : R) (Hx : -1 <= x <= 1) :
   {l : R | Un_cv (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}.x:R
Hx:-1 <= x <= 1
{l : R |
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}
Proof.x:R
Hx:-1 <= x <= 1
{l : R |
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}
destruct (Rle_lt_dec 0 x).x:R
Hx:-1 <= x <= 1
r:0 <= x
{l : R |
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}
x:R
Hx:-1 <= x <= 1
r:x < 0
{l : R |
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}
 assert (pr : 0 <= x <= 1) by tauto.x:R
Hx:-1 <= x <= 1
r:0 <= x
pr:0 <= x <= 1
{l : R |
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}
x:R
Hx:-1 <= x <= 1
r:x < 0
{l : R |
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}
 exact (ps_atan_exists_01 x pr).x:R
Hx:-1 <= x <= 1
r:x < 0
{l : R |
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}
assert (pr : 0 <= -x <= 1) by (destruct Hx; split; lra).x:R
Hx:-1 <= x <= 1
r:x < 0
pr:0 <= - x <= 1
{l : R |
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}
destruct (ps_atan_exists_01 _ pr) as [v Pv].x:R
Hx:-1 <= x <= 1
r:x < 0
pr:0 <= - x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
{l : R |
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}
exists (-v).x:R
Hx:-1 <= x <= 1
r:x < 0
pr:0 <= - x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N)
  (- v)
 apply (Un_cv_ext (fun n => (- 1) * sum_f_R0 (tg_alt (Ratan_seq (- x))) n)).x:R
Hx:-1 <= x <= 1
r:x < 0
pr:0 <= - x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
forall n : nat,
-1 * sum_f_R0 (tg_alt (Ratan_seq (- x))) n =
sum_f_R0 (tg_alt (Ratan_seq x)) n
x:R
Hx:-1 <= x <= 1
r:x < 0
pr:0 <= - x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
Un_cv
  (fun n : nat =>
   -1 * sum_f_R0 (tg_alt (Ratan_seq (- x))) n) (- v)
 intros n; rewrite sum_Ratan_seq_opp; ring.x:R
Hx:-1 <= x <= 1
r:x < 0
pr:0 <= - x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
Un_cv
  (fun n : nat =>
   -1 * sum_f_R0 (tg_alt (Ratan_seq (- x))) n) (- v)
replace (-v) with (-1 * v) by ring.x:R
Hx:-1 <= x <= 1
r:x < 0
pr:0 <= - x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
Un_cv
  (fun n : nat =>
   -1 * sum_f_R0 (tg_alt (Ratan_seq (- x))) n)
  (-1 * v)
apply CV_mult;[ | assumption].x:R
Hx:-1 <= x <= 1
r:x < 0
pr:0 <= - x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
Un_cv (fun _ : nat => -1) (-1)
solve[intros; exists 0%nat; intros; rewrite R_dist_eq; auto].
Qed.

Definition in_int (x : R) : {-1 <= x <= 1}+{~ -1 <= x <= 1}.x:R
{-1 <= x <= 1} + {~ -1 <= x <= 1}
Proof.x:R
{-1 <= x <= 1} + {~ -1 <= x <= 1}
destruct (Rle_lt_dec x 1).x:R
r:x <= 1
{-1 <= x <= 1} + {~ -1 <= x <= 1}
x:R
r:1 < x
{-1 <= x <= 1} + {~ -1 <= x <= 1}
 destruct (Rle_lt_dec (-1) x).x:R
r:x <= 1
r0:-1 <= x
{-1 <= x <= 1} + {~ -1 <= x <= 1}
x:R
r:x <= 1
r0:x < -1
{-1 <= x <= 1} + {~ -1 <= x <= 1}
x:R
r:1 < x
{-1 <= x <= 1} + {~ -1 <= x <= 1}
  left;split; auto.x:R
r:x <= 1
r0:x < -1
{-1 <= x <= 1} + {~ -1 <= x <= 1}
x:R
r:1 < x
{-1 <= x <= 1} + {~ -1 <= x <= 1}
 right;intros [a1 a2]; lra.x:R
r:1 < x
{-1 <= x <= 1} + {~ -1 <= x <= 1}
right;intros [a1 a2]; lra.
Qed.

Definition ps_atan (x : R) : R :=
 match in_int x with
   left h => let (v, _) := ps_atan_exists_1 x h in v
 | right h => atan x
 end.

Proof of the equivalence of the two definitions between -1 and 1

Lemma ps_atan0_0 : ps_atan 0 = 0.ps_atan 0 = 0
Proof.ps_atan 0 = 0
unfold ps_atan.match in_int 0 with
| left h => let (v, _) := ps_atan_exists_1 0 h in v
| right _ => atan 0
end = 0
 destruct (in_int 0) as [h1 | h2].h1:-1 <= 0 <= 1
(let (v, _) := ps_atan_exists_1 0 h1 in v) = 0
h2:~ -1 <= 0 <= 1
atan 0 = 0
 destruct (ps_atan_exists_1 0 h1) as [v P].h1:-1 <= 0 <= 1
v:R
P:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq 0)) N) v
v = 0
h2:~ -1 <= 0 <= 1
atan 0 = 0
 apply (UL_sequence _ _ _ P).h1:-1 <= 0 <= 1
v:R
P:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq 0)) N) v
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq 0)) N) 0
h2:~ -1 <= 0 <= 1
atan 0 = 0
  apply (Un_cv_ext (fun n => 0)).h1:-1 <= 0 <= 1
v:R
P:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq 0)) N) v
forall n : nat, 0 = sum_f_R0 (tg_alt (Ratan_seq 0)) n
h1:-1 <= 0 <= 1
v:R
P:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq 0)) N) v
Un_cv (fun _ : nat => 0) 0
h2:~ -1 <= 0 <= 1
atan 0 = 0
  symmetry;apply sum_eq_R0.h1:-1 <= 0 <= 1
v:R
P:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq 0)) N) v
n:nat
forall n0 : nat,
(n0 <= n)%nat -> tg_alt (Ratan_seq 0) n0 = 0
h1:-1 <= 0 <= 1
v:R
P:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq 0)) N) v
Un_cv (fun _ : nat => 0) 0
h2:~ -1 <= 0 <= 1
atan 0 = 0
  intros i _; unfold tg_alt, Ratan_seq; rewrite plus_comm; simpl.h1:-1 <= 0 <= 1
v:R
P:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq 0)) N) v
n, i:nat
(-1) ^ i *
(0 * 0 ^ (i + (i + 0)) /
 match (i + (i + 0))%nat with
 | 0%nat => 1
 | S _ => INR (i + (i + 0)) + 1
 end) = 0
h1:-1 <= 0 <= 1
v:R
P:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq 0)) N) v
Un_cv (fun _ : nat => 0) 0
h2:~ -1 <= 0 <= 1
atan 0 = 0
  unfold Rdiv; rewrite !Rmult_0_l, Rmult_0_r; reflexivity.h1:-1 <= 0 <= 1
v:R
P:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq 0)) N) v
Un_cv (fun _ : nat => 0) 0
h2:~ -1 <= 0 <= 1
atan 0 = 0
 intros eps ep; exists 0%nat; intros n _; unfold R_dist.h1:-1 <= 0 <= 1
v:R
P:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq 0)) N) v
eps:R
ep:eps > 0
n:nat
Rabs (0 - 0) < eps
h2:~ -1 <= 0 <= 1
atan 0 = 0
 rewrite Rminus_0_r, Rabs_pos_eq; auto with real.h2:~ -1 <= 0 <= 1
atan 0 = 0
case h2; split; lra.
Qed.

Lemma ps_atan_exists_1_opp :
  forall x h h', proj1_sig (ps_atan_exists_1 (-x) h) = 
     -(proj1_sig (ps_atan_exists_1 x h')).forall (x : R) (h : -1 <= - x <= 1)
  (h' : -1 <= x <= 1),
proj1_sig (ps_atan_exists_1 (- x) h) =
- proj1_sig (ps_atan_exists_1 x h')
Proof.forall (x : R) (h : -1 <= - x <= 1)
  (h' : -1 <= x <= 1),
proj1_sig (ps_atan_exists_1 (- x) h) =
- proj1_sig (ps_atan_exists_1 x h')
intros x h h'; destruct (ps_atan_exists_1 (-x) h) as [v Pv].x:R
h:-1 <= - x <= 1
h':-1 <= x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
proj1_sig
  (exist
     (fun l : R =>
      Un_cv
        (fun N : nat =>
         sum_f_R0 (tg_alt (Ratan_seq (- x))) N) l) v
     Pv) = - proj1_sig (ps_atan_exists_1 x h')
destruct (ps_atan_exists_1 x h') as [u Pu]; simpl.x:R
h:-1 <= - x <= 1
h':-1 <= x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
u:R
Pu:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) u
v = - u
assert (Pu' : Un_cv (fun N => (-1) * sum_f_R0 (tg_alt (Ratan_seq x)) N) (-1 * u)).x:R
h:-1 <= - x <= 1
h':-1 <= x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
u:R
Pu:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) u
Un_cv
  (fun N : nat =>
   -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) (-1 * u)
x:R
h:-1 <= - x <= 1
h':-1 <= x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
u:R
Pu:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) u
Pu':Un_cv (fun N : nat => -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) (-1 * u)
v = - u
 apply CV_mult;[ | assumption].x:R
h:-1 <= - x <= 1
h':-1 <= x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
u:R
Pu:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) u
Un_cv (fun _ : nat => -1) (-1)
x:R
h:-1 <= - x <= 1
h':-1 <= x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
u:R
Pu:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) u
Pu':Un_cv (fun N : nat => -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) (-1 * u)
v = - u
 intros eps ep; exists 0%nat; intros; rewrite R_dist_eq; assumption.x:R
h:-1 <= - x <= 1
h':-1 <= x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
u:R
Pu:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) u
Pu':Un_cv (fun N : nat => -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) (-1 * u)
v = - u  
assert (Pv' : Un_cv
           (fun N : nat => -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) v).x:R
h:-1 <= - x <= 1
h':-1 <= x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
u:R
Pu:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) u
Pu':Un_cv (fun N : nat => -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) (-1 * u)
Un_cv
  (fun N : nat =>
   -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) v
x:R
h:-1 <= - x <= 1
h':-1 <= x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
u:R
Pu:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) u
Pu':Un_cv (fun N : nat => -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) (-1 * u)
Pv':Un_cv (fun N : nat => -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) v
v = - u
 apply Un_cv_ext with (2 := Pv); intros n; rewrite sum_Ratan_seq_opp; ring.x:R
h:-1 <= - x <= 1
h':-1 <= x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
u:R
Pu:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) u
Pu':Un_cv (fun N : nat => -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) (-1 * u)
Pv':Un_cv (fun N : nat => -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) v
v = - u
replace (-u) with (-1 * u) by ring.x:R
h:-1 <= - x <= 1
h':-1 <= x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq (- x))) N) v
u:R
Pu:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) u
Pu':Un_cv (fun N : nat => -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) (-1 * u)
Pv':Un_cv (fun N : nat => -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) v
v = -1 * u
apply UL_sequence with (1:=Pv') (2:= Pu').
Qed.

Lemma ps_atan_opp : forall x, ps_atan (-x) = -ps_atan x.forall x : R, ps_atan (- x) = - ps_atan x
Proof.forall x : R, ps_atan (- x) = - ps_atan x
intros x; unfold ps_atan.x:R
match in_int (- x) with
| left h =>
    let (v, _) := ps_atan_exists_1 (- x) h in v
| right _ => atan (- x)
end =
-
match in_int x with
| left h => let (v, _) := ps_atan_exists_1 x h in v
| right _ => atan x
end
destruct (in_int (- x)) as [inside | outside].x:R
inside:-1 <= - x <= 1
(let (v, _) := ps_atan_exists_1 (- x) inside in v) =
-
match in_int x with
| left h => let (v, _) := ps_atan_exists_1 x h in v
| right _ => atan x
end
x:R
outside:~ -1 <= - x <= 1
atan (- x) =
-
match in_int x with
| left h => let (v, _) := ps_atan_exists_1 x h in v
| right _ => atan x
end
 destruct (in_int x) as [ins' | outs'].x:R
inside:-1 <= - x <= 1
ins':-1 <= x <= 1
(let (v, _) := ps_atan_exists_1 (- x) inside in v) =
- (let (v, _) := ps_atan_exists_1 x ins' in v)
x:R
inside:-1 <= - x <= 1
outs':~ -1 <= x <= 1
(let (v, _) := ps_atan_exists_1 (- x) inside in v) =
- atan x
x:R
outside:~ -1 <= - x <= 1
atan (- x) =
-
match in_int x with
| left h => let (v, _) := ps_atan_exists_1 x h in v
| right _ => atan x
end
 generalize (ps_atan_exists_1_opp x inside ins').x:R
inside:-1 <= - x <= 1
ins':-1 <= x <= 1
proj1_sig (ps_atan_exists_1 (- x) inside) =
- proj1_sig (ps_atan_exists_1 x ins') ->
(let (v, _) := ps_atan_exists_1 (- x) inside in v) =
- (let (v, _) := ps_atan_exists_1 x ins' in v)
x:R
inside:-1 <= - x <= 1
outs':~ -1 <= x <= 1
(let (v, _) := ps_atan_exists_1 (- x) inside in v) =
- atan x
x:R
outside:~ -1 <= - x <= 1
atan (- x) =
-
match in_int x with
| left h => let (v, _) := ps_atan_exists_1 x h in v
| right _ => atan x
end
  intros h; exact h.x:R
inside:-1 <= - x <= 1
outs':~ -1 <= x <= 1
(let (v, _) := ps_atan_exists_1 (- x) inside in v) =
- atan x
x:R
outside:~ -1 <= - x <= 1
atan (- x) =
-
match in_int x with
| left h => let (v, _) := ps_atan_exists_1 x h in v
| right _ => atan x
end
 destruct inside; case outs'; split; lra.x:R
outside:~ -1 <= - x <= 1
atan (- x) =
-
match in_int x with
| left h => let (v, _) := ps_atan_exists_1 x h in v
| right _ => atan x
end
destruct (in_int x) as [ins' | outs'].x:R
outside:~ -1 <= - x <= 1
ins':-1 <= x <= 1
atan (- x) =
- (let (v, _) := ps_atan_exists_1 x ins' in v)
x:R
outside:~ -1 <= - x <= 1
outs':~ -1 <= x <= 1
atan (- x) = - atan x
 destruct outside; case ins'; split; lra.x:R
outside:~ -1 <= - x <= 1
outs':~ -1 <= x <= 1
atan (- x) = - atan x
apply atan_opp.
Qed.

atan = ps_atan

Lemma ps_atanSeq_continuity_pt_1 : forall (N:nat) (x:R),
      0 <= x ->
      x <= 1 ->
      continuity_pt (fun x => sum_f_R0 (tg_alt (Ratan_seq x)) N) x.forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
Proof.forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
assert (Sublemma : forall (x:R) (N:nat), sum_f_R0 (tg_alt (Ratan_seq x)) N = x * (comp (fun x => sum_f_R0 (fun n => (fun i : nat => (-1) ^ i / INR (2 * i + 1)) n * x ^ n) N) (fun x => x ^ 2) x)).forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
 intros x N.x:R
N:nat
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  induction N.x:R
sum_f_R0 (tg_alt (Ratan_seq x)) 0 =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 0)
  (fun x0 : R => x0 ^ 2) x
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
sum_f_R0 (tg_alt (Ratan_seq x)) (S N) =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2) x
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   unfold tg_alt, Ratan_seq, comp ; simpl ; field.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
sum_f_R0 (tg_alt (Ratan_seq x)) (S N) =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2) x
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   simpl sum_f_R0 at 1.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
sum_f_R0 (tg_alt (Ratan_seq x)) N +
tg_alt (Ratan_seq x) (S N) =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2) x
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   rewrite IHN.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x +
tg_alt (Ratan_seq x) (S N) =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2) x
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   replace (comp (fun x => sum_f_R0 (fun n : nat => (-1) ^ n / INR (2 * n + 1) * x ^ n) (S N)) (fun x => x ^ 2))
     with (comp (fun x => sum_f_R0 (fun n : nat => (-1) ^ n / INR (2 * n + 1) * x ^ n) N + (-1) ^ (S N) / INR (2 * (S N) + 1) * x ^ (S N)) (fun x => x ^ 2)).x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x +
tg_alt (Ratan_seq x) (S N) =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) x
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   unfold comp.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
x *
sum_f_R0
  (fun n : nat =>
   (-1) ^ n / INR (2 * n + 1) * (x ^ 2) ^ n) N +
tg_alt (Ratan_seq x) (S N) =
x *
(sum_f_R0
   (fun n : nat =>
    (-1) ^ n / INR (2 * n + 1) * (x ^ 2) ^ n) N +
 (-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   rewrite Rmult_plus_distr_l.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
x *
sum_f_R0
  (fun n : nat =>
   (-1) ^ n / INR (2 * n + 1) * (x ^ 2) ^ n) N +
tg_alt (Ratan_seq x) (S N) =
x *
sum_f_R0
  (fun n : nat =>
   (-1) ^ n / INR (2 * n + 1) * (x ^ 2) ^ n) N +
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   apply Rplus_eq_compat_l.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
tg_alt (Ratan_seq x) (S N) =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   unfold tg_alt, Ratan_seq.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
(-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   rewrite <- Rmult_assoc.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
(-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   case (Req_dec x 0) ; intro Hyp.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x = 0
(-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   rewrite Hyp ; rewrite pow_i.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x = 0
(-1) ^ S N * (0 / INR (2 * S N + 1)) =
0 * ((-1) ^ S N / INR (2 * S N + 1)) * (0 ^ 2) ^ S N
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x = 0
(0 < 2 * S N + 1)%nat
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x rewrite Rmult_0_l ; rewrite Rmult_0_l.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x = 0
(-1) ^ S N * (0 / INR (2 * S N + 1)) = 0
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x = 0
(0 < 2 * S N + 1)%nat
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   unfold Rdiv ; rewrite Rmult_0_l ; rewrite Rmult_0_r ; reflexivity.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x = 0
(0 < 2 * S N + 1)%nat
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   intuition.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   replace (x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N) with (x ^ (2 * S N + 1) * ((-1) ^ S N / INR (2 * S N + 1))).x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1)) =
x ^ (2 * S N + 1) * ((-1) ^ S N / INR (2 * S N + 1))
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
x ^ (2 * S N + 1) * ((-1) ^ S N / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   rewrite Rmult_comm ; unfold Rdiv at 1.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
x ^ (2 * S N + 1) * / INR (2 * S N + 1) * (-1) ^ S N =
x ^ (2 * S N + 1) * ((-1) ^ S N / INR (2 * S N + 1))
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
x ^ (2 * S N + 1) * ((-1) ^ S N / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   rewrite Rmult_assoc ; apply Rmult_eq_compat_l.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
/ INR (2 * S N + 1) * (-1) ^ S N =
(-1) ^ S N / INR (2 * S N + 1)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
x ^ (2 * S N + 1) * ((-1) ^ S N / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   field.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
INR (2 * S N + 1) <> 0
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
x ^ (2 * S N + 1) * ((-1) ^ S N / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x apply Rgt_not_eq ; intuition.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
x ^ (2 * S N + 1) * ((-1) ^ S N / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   rewrite Rmult_assoc.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
x ^ (2 * S N + 1) * ((-1) ^ S N / INR (2 * S N + 1)) =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   replace (x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)) with (((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N) * x).x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
x ^ (2 * S N + 1) * ((-1) ^ S N / INR (2 * S N + 1)) =
(-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N * x
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N * x =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   rewrite Rmult_assoc.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
x ^ (2 * S N + 1) * ((-1) ^ S N / INR (2 * S N + 1)) =
(-1) ^ S N / INR (2 * S N + 1) * ((x ^ 2) ^ S N * x)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N * x =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   replace ((x ^ 2) ^ S N * x) with (x ^ (2 * S N + 1)).x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
x ^ (2 * S N + 1) * ((-1) ^ S N / INR (2 * S N + 1)) =
(-1) ^ S N / INR (2 * S N + 1) * x ^ (2 * S N + 1)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
x ^ (2 * S N + 1) = (x ^ 2) ^ S N * x
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N * x =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   rewrite Rmult_comm at 1 ; reflexivity.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
x ^ (2 * S N + 1) = (x ^ 2) ^ S N * x
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N * x =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   rewrite <- pow_mult.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
x ^ (2 * S N + 1) = x ^ (2 * S N) * x
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N * x =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   assert (Temp : forall x n, x ^ n * x = x ^ (n+1)).x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
forall (x0 : R) (n : nat), x0 ^ n * x0 = x0 ^ (n + 1)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
Temp:forall (x0 : R) (n : nat),
x0 ^ n * x0 = x0 ^ (n + 1)
x ^ (2 * S N + 1) = x ^ (2 * S N) * x
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N * x =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
    intros a n ; induction n.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
a:R
a ^ 0 * a = a ^ (0 + 1)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n0 : nat =>
      (-1) ^ n0 / INR (2 * n0 + 1) * x0 ^ n0) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
a:R
n:nat
IHn:a ^ n * a = a ^ (n + 1)
a ^ S n * a = a ^ (S n + 1)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
Temp:forall (x0 : R) (n : nat),
x0 ^ n * x0 = x0 ^ (n + 1)
x ^ (2 * S N + 1) = x ^ (2 * S N) * x
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N * x =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x rewrite pow_O.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
a:R
1 * a = a ^ (0 + 1)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n0 : nat =>
      (-1) ^ n0 / INR (2 * n0 + 1) * x0 ^ n0) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
a:R
n:nat
IHn:a ^ n * a = a ^ (n + 1)
a ^ S n * a = a ^ (S n + 1)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
Temp:forall (x0 : R) (n : nat),
x0 ^ n * x0 = x0 ^ (n + 1)
x ^ (2 * S N + 1) = x ^ (2 * S N) * x
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N * x =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x simpl ; intuition.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n0 : nat =>
      (-1) ^ n0 / INR (2 * n0 + 1) * x0 ^ n0) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
a:R
n:nat
IHn:a ^ n * a = a ^ (n + 1)
a ^ S n * a = a ^ (S n + 1)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
Temp:forall (x0 : R) (n : nat),
x0 ^ n * x0 = x0 ^ (n + 1)
x ^ (2 * S N + 1) = x ^ (2 * S N) * x
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N * x =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
    simpl ; rewrite Rmult_assoc ; rewrite IHn ; intuition.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
Temp:forall (x0 : R) (n : nat),
x0 ^ n * x0 = x0 ^ (n + 1)
x ^ (2 * S N + 1) = x ^ (2 * S N) * x
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N * x =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   rewrite Temp ; reflexivity.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
Hyp:x <> 0
(-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N * x =
x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)
x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   rewrite Rmult_comm ; reflexivity.x:R
N:nat
IHN:sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N)
  (fun x0 : R => x0 ^ 2) x
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) N +
   (-1) ^ S N / INR (2 * S N + 1) * x0 ^ S N)
  (fun x0 : R => x0 ^ 2) =
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x0 ^ n) 
     (S N)) (fun x0 : R => x0 ^ 2)
Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
   intuition.Sublemma:forall (x : R) (N : nat),
sum_f_R0 (tg_alt (Ratan_seq x)) N =
x *
comp
  (fun x0 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x0 ^ n) N) (fun x0 : R => x0 ^ 2) x
forall (N : nat) (x : R),
0 <= x ->
x <= 1 ->
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
intros N x x_lb x_ub.Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
continuity_pt
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
 intros eps eps_pos.Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
     (sum_f_R0 (tg_alt (Ratan_seq x)) N) < eps)
 assert (continuity_id : continuity id).Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity id
Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
     (sum_f_R0 (tg_alt (Ratan_seq x)) N) < eps)
 apply derivable_continuous ; exact derivable_id.Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
     (sum_f_R0 (tg_alt (Ratan_seq x)) N) < eps)
assert (Temp := continuity_mult id (comp
        (fun x1 : R =>
         sum_f_R0 (fun n : nat => (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
        (fun x1 : R => x1 ^ 2))
           continuity_id).Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
Temp:continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2)) ->
continuity
  (id *
   comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2))
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
     (sum_f_R0 (tg_alt (Ratan_seq x)) N) < eps)
assert (Temp2 : continuity
    (comp
       (fun x1 : R =>
        sum_f_R0 (fun n : nat => (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
       (fun x1 : R => x1 ^ 2))).Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
Temp:continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2)) ->
continuity
  (id *
   comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2))
continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2))
Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
Temp:continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2)) ->
continuity
  (id *
   comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2))
Temp2:continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n)
        N) (fun x1 : R => x1 ^ 2))
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
     (sum_f_R0 (tg_alt (Ratan_seq x)) N) < eps)
 apply continuity_comp.Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
Temp:continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2)) ->
continuity
  (id *
   comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2))
continuity (fun x1 : R => x1 ^ 2)
Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
Temp:continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2)) ->
continuity
  (id *
   comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2))
continuity
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
Temp:continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2)) ->
continuity
  (id *
   comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2))
Temp2:continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n)
        N) (fun x1 : R => x1 ^ 2))
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
     (sum_f_R0 (tg_alt (Ratan_seq x)) N) < eps)
 reg.Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
Temp:continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2)) ->
continuity
  (id *
   comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2))
continuity
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
Temp:continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2)) ->
continuity
  (id *
   comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2))
Temp2:continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n)
        N) (fun x1 : R => x1 ^ 2))
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
     (sum_f_R0 (tg_alt (Ratan_seq x)) N) < eps)
 apply continuity_finite_sum.Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
Temp:continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2)) ->
continuity
  (id *
   comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2))
Temp2:continuity
  (comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n)
        N) (fun x1 : R => x1 ^ 2))
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
     (sum_f_R0 (tg_alt (Ratan_seq x)) N) < eps)
 elim (Temp Temp2 x eps eps_pos) ; clear Temp Temp2 ; intros alpha T ; destruct T as (alpha_pos, T).Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
alpha:R
alpha_pos:alpha > 0
T:forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < alpha ->
dist R_met
  ((id *
    comp
      (fun x1 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x1 ^ n)%R)
         N) (fun x1 : R => x1 ^ 2))%F x0)
  ((id *
    comp
      (fun x1 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x1 ^ n)%R)
         N) (fun x1 : R => x1 ^ 2))%F x) < eps
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
     (sum_f_R0 (tg_alt (Ratan_seq x)) N) < eps)
 exists alpha ; split.Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
alpha:R
alpha_pos:alpha > 0
T:forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < alpha ->
dist R_met
  ((id *
    comp
      (fun x1 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x1 ^ n)%R)
         N) (fun x1 : R => x1 ^ 2))%F x0)
  ((id *
    comp
      (fun x1 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x1 ^ n)%R)
         N) (fun x1 : R => x1 ^ 2))%F x) < eps
alpha > 0
Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
alpha:R
alpha_pos:alpha > 0
T:forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < alpha ->
dist R_met
  ((id *
    comp
      (fun x1 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x1 ^ n)%R)
         N) (fun x1 : R => x1 ^ 2))%F x0)
  ((id *
    comp
      (fun x1 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x1 ^ n)%R)
         N) (fun x1 : R => x1 ^ 2))%F x) < eps
forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < alpha ->
dist R_met (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq x)) N) < eps
 intuition.Sublemma:forall (x0 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x0)) N0 =
x0 *
comp
  (fun x1 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x1 ^ n) N0) (fun x1 : R => x1 ^ 2) x0
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
alpha:R
alpha_pos:alpha > 0
T:forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < alpha ->
dist R_met
  ((id *
    comp
      (fun x1 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x1 ^ n)%R)
         N) (fun x1 : R => x1 ^ 2))%F x0)
  ((id *
    comp
      (fun x1 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x1 ^ n)%R)
         N) (fun x1 : R => x1 ^ 2))%F x) < eps
forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < alpha ->
dist R_met (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq x)) N) < eps
intros x0 x0_cond.Sublemma:forall (x1 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x1)) N0 =
x1 *
comp
  (fun x2 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x2 ^ n) N0) (fun x2 : R => x2 ^ 2) x1
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
alpha:R
alpha_pos:alpha > 0
T:forall x1 : Base R_met,
D_x no_cond x x1 /\ dist R_met x1 x < alpha ->
dist R_met
  ((id *
    comp
      (fun x2 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x2 ^ n)%R)
         N) (fun x2 : R => x2 ^ 2))%F x1)
  ((id *
    comp
      (fun x2 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x2 ^ n)%R)
         N) (fun x2 : R => x2 ^ 2))%F x) < eps
x0:Base R_met
x0_cond:D_x no_cond x x0 /\ dist R_met x0 x < alpha
dist R_met (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq x)) N) < eps
 rewrite Sublemma ; rewrite Sublemma.Sublemma:forall (x1 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x1)) N0 =
x1 *
comp
  (fun x2 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x2 ^ n) N0) (fun x2 : R => x2 ^ 2) x1
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
alpha:R
alpha_pos:alpha > 0
T:forall x1 : Base R_met,
D_x no_cond x x1 /\ dist R_met x1 x < alpha ->
dist R_met
  ((id *
    comp
      (fun x2 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x2 ^ n)%R)
         N) (fun x2 : R => x2 ^ 2))%F x1)
  ((id *
    comp
      (fun x2 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x2 ^ n)%R)
         N) (fun x2 : R => x2 ^ 2))%F x) < eps
x0:Base R_met
x0_cond:D_x no_cond x x0 /\ dist R_met x0 x < alpha
dist R_met
  (x0 *
   comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2) x0)
  (x *
   comp
     (fun x1 : R =>
      sum_f_R0
        (fun n : nat =>
         (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
     (fun x1 : R => x1 ^ 2) x) < eps
apply T.Sublemma:forall (x1 : R) (N0 : nat),
sum_f_R0 (tg_alt (Ratan_seq x1)) N0 =
x1 *
comp
  (fun x2 : R =>
   sum_f_R0
     (fun n : nat =>
      (fun i : nat =>
       (-1) ^ i / INR (2 * i + 1)) n *
      x2 ^ n) N0) (fun x2 : R => x2 ^ 2) x1
N:nat
x:R
x_lb:0 <= x
x_ub:x <= 1
eps:R
eps_pos:eps > 0
continuity_id:continuity id
alpha:R
alpha_pos:alpha > 0
T:forall x1 : Base R_met,
D_x no_cond x x1 /\ dist R_met x1 x < alpha ->
dist R_met
  ((id *
    comp
      (fun x2 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x2 ^ n)%R)
         N) (fun x2 : R => x2 ^ 2))%F x1)
  ((id *
    comp
      (fun x2 : R =>
       sum_f_R0
         (fun n : nat =>
          ((-1) ^ n / INR (2 * n + 1) * x2 ^ n)%R)
         N) (fun x2 : R => x2 ^ 2))%F x) < eps
x0:Base R_met
x0_cond:D_x no_cond x x0 /\ dist R_met x0 x < alpha
D_x no_cond x x0 /\ dist R_met x0 x < alpha
intuition.
Qed.

Definition of ps_atan's derivative

Definition Datan_seq := fun (x:R) (n:nat) => x ^ (2*n).

Lemma pow_lt_1_compat : forall x n, 0 <= x < 1 -> (0 < n)%nat ->
   0 <= x ^ n < 1.forall (x : R) (n : nat),
0 <= x < 1 -> (0 < n)%nat -> 0 <= x ^ n < 1
Proof.forall (x : R) (n : nat),
0 <= x < 1 -> (0 < n)%nat -> 0 <= x ^ n < 1
intros x n hx; induction 1; simpl.x:R
hx:0 <= x < 1
0 <= x * 1 < 1
x:R
hx:0 <= x < 1
m:nat
H:(1 <= m)%nat
IHle:0 <= x ^ m < 1
0 <= x * x ^ m < 1
 rewrite Rmult_1_r; tauto.x:R
hx:0 <= x < 1
m:nat
H:(1 <= m)%nat
IHle:0 <= x ^ m < 1
0 <= x * x ^ m < 1
split.x:R
hx:0 <= x < 1
m:nat
H:(1 <= m)%nat
IHle:0 <= x ^ m < 1
0 <= x * x ^ m
x:R
hx:0 <= x < 1
m:nat
H:(1 <= m)%nat
IHle:0 <= x ^ m < 1
x * x ^ m < 1
 apply Rmult_le_pos; tauto.x:R
hx:0 <= x < 1
m:nat
H:(1 <= m)%nat
IHle:0 <= x ^ m < 1
x * x ^ m < 1
rewrite <- (Rmult_1_r 1); apply Rmult_le_0_lt_compat; intuition.
Qed.

Lemma Datan_seq_Rabs : forall x n, Datan_seq (Rabs x) n = Datan_seq x n.forall (x : R) (n : nat),
Datan_seq (Rabs x) n = Datan_seq x n
Proof.forall (x : R) (n : nat),
Datan_seq (Rabs x) n = Datan_seq x n
intros x n; unfold Datan_seq; rewrite !pow_mult, pow2_abs; reflexivity.
Qed.

Lemma Datan_seq_pos : forall x n, 0 < x -> 0 < Datan_seq x n.forall (x : R) (n : nat), 0 < x -> 0 < Datan_seq x n
Proof.forall (x : R) (n : nat), 0 < x -> 0 < Datan_seq x n
intros x n x_lb ; unfold Datan_seq ; induction n.x:R
x_lb:0 < x
0 < x ^ (2 * 0)
x:R
n:nat
x_lb:0 < x
IHn:0 < x ^ (2 * n)
0 < x ^ (2 * S n)
 simpl ; intuition.x:R
n:nat
x_lb:0 < x
IHn:0 < x ^ (2 * n)
0 < x ^ (2 * S n)
 replace (x ^ (2 * S n)) with ((x ^ 2) * (x ^ (2 * n))).x:R
n:nat
x_lb:0 < x
IHn:0 < x ^ (2 * n)
0 < x ^ 2 * x ^ (2 * n)
x:R
n:nat
x_lb:0 < x
IHn:0 < x ^ (2 * n)
x ^ 2 * x ^ (2 * n) = x ^ (2 * S n)
 apply Rmult_gt_0_compat.x:R
n:nat
x_lb:0 < x
IHn:0 < x ^ (2 * n)
x ^ 2 > 0
x:R
n:nat
x_lb:0 < x
IHn:0 < x ^ (2 * n)
x ^ (2 * n) > 0
x:R
n:nat
x_lb:0 < x
IHn:0 < x ^ (2 * n)
x ^ 2 * x ^ (2 * n) = x ^ (2 * S n)
 replace (x^2) with (x*x) by field ; apply Rmult_gt_0_compat ; assumption.x:R
n:nat
x_lb:0 < x
IHn:0 < x ^ (2 * n)
x ^ (2 * n) > 0
x:R
n:nat
x_lb:0 < x
IHn:0 < x ^ (2 * n)
x ^ 2 * x ^ (2 * n) = x ^ (2 * S n)
 assumption.x:R
n:nat
x_lb:0 < x
IHn:0 < x ^ (2 * n)
x ^ 2 * x ^ (2 * n) = x ^ (2 * S n)
 replace (2 * S n)%nat with (S (S (2 * n))) by intuition.x:R
n:nat
x_lb:0 < x
IHn:0 < x ^ (2 * n)
x ^ 2 * x ^ (2 * n) = x ^ S (S (2 * n))
 simpl ; field.
Qed.

Lemma Datan_sum_eq :forall x n,
  sum_f_R0 (tg_alt (Datan_seq x)) n = (1 - (- x ^ 2) ^ S n)/(1 + x ^ 2).forall (x : R) (n : nat),
sum_f_R0 (tg_alt (Datan_seq x)) n =
(1 - (- x ^ 2) ^ S n) / (1 + x ^ 2)
Proof.forall (x : R) (n : nat),
sum_f_R0 (tg_alt (Datan_seq x)) n =
(1 - (- x ^ 2) ^ S n) / (1 + x ^ 2)
intros x n.x:R
n:nat
sum_f_R0 (tg_alt (Datan_seq x)) n =
(1 - (- x ^ 2) ^ S n) / (1 + x ^ 2)
assert (dif : - x ^ 2 <> 1).x:R
n:nat
- x ^ 2 <> 1
x:R
n:nat
dif:- x ^ 2 <> 1
sum_f_R0 (tg_alt (Datan_seq x)) n =
(1 - (- x ^ 2) ^ S n) / (1 + x ^ 2)
apply Rlt_not_eq; apply Rle_lt_trans with 0;[ | apply Rlt_0_1].x:R
n:nat
- x ^ 2 <= 0
x:R
n:nat
dif:- x ^ 2 <> 1
sum_f_R0 (tg_alt (Datan_seq x)) n =
(1 - (- x ^ 2) ^ S n) / (1 + x ^ 2)
assert (t := pow2_ge_0 x); lra.x:R
n:nat
dif:- x ^ 2 <> 1
sum_f_R0 (tg_alt (Datan_seq x)) n =
(1 - (- x ^ 2) ^ S n) / (1 + x ^ 2)
replace (1 + x ^ 2) with (1 - - (x ^ 2)) by ring; rewrite <- (tech3 _ n dif).x:R
n:nat
dif:- x ^ 2 <> 1
sum_f_R0 (tg_alt (Datan_seq x)) n =
sum_f_R0 (fun i : nat => (- x ^ 2) ^ i) n
apply sum_eq; unfold tg_alt, Datan_seq; intros i _.x:R
n:nat
dif:- x ^ 2 <> 1
i:nat
(-1) ^ i * x ^ (2 * i) = (- x ^ 2) ^ i
rewrite pow_mult, <- Rpow_mult_distr.x:R
n:nat
dif:- x ^ 2 <> 1
i:nat
(-1 * x ^ 2) ^ i = (- x ^ 2) ^ i
f_equal.x:R
n:nat
dif:- x ^ 2 <> 1
i:nat
-1 * x ^ 2 = - x ^ 2
ring.
Qed.

Lemma Datan_seq_increasing : forall x y n, (n > 0)%nat -> 0 <= x < y -> Datan_seq x n < Datan_seq y n.forall (x y : R) (n : nat),
(n > 0)%nat ->
0 <= x < y -> Datan_seq x n < Datan_seq y n
Proof.forall (x y : R) (n : nat),
(n > 0)%nat ->
0 <= x < y -> Datan_seq x n < Datan_seq y n
intros x y n n_lb x_encad ; assert (x_pos : x >= 0) by intuition.x, y:R
n:nat
n_lb:(n > 0)%nat
x_encad:0 <= x < y
x_pos:x >= 0
Datan_seq x n < Datan_seq y n
 assert (y_pos : y > 0).x, y:R
n:nat
n_lb:(n > 0)%nat
x_encad:0 <= x < y
x_pos:x >= 0
y > 0
x, y:R
n:nat
n_lb:(n > 0)%nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
Datan_seq x n < Datan_seq y n apply Rle_lt_trans with (r2:=x) ; intuition.x, y:R
n:nat
n_lb:(n > 0)%nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
Datan_seq x n < Datan_seq y n
 induction n.x, y:R
n_lb:(0 > 0)%nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
Datan_seq x 0 < Datan_seq y 0
x, y:R
n:nat
n_lb:(S n > 0)%nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:(n > 0)%nat -> Datan_seq x n < Datan_seq y n
Datan_seq x (S n) < Datan_seq y (S n)
 apply False_ind ; intuition.x, y:R
n:nat
n_lb:(S n > 0)%nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:(n > 0)%nat -> Datan_seq x n < Datan_seq y n
Datan_seq x (S n) < Datan_seq y (S n)
 clear -x_encad x_pos y_pos ; induction n ; unfold Datan_seq.x, y:R
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
x ^ (2 * 1) < y ^ (2 * 1)
x, y:R
n:nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
 case x_pos ; clear x_pos ; intro x_pos.x, y:R
x_encad:0 <= x < y
y_pos:y > 0
x_pos:x > 0
x ^ (2 * 1) < y ^ (2 * 1)
x, y:R
x_encad:0 <= x < y
y_pos:y > 0
x_pos:x = 0
x ^ (2 * 1) < y ^ (2 * 1)
x, y:R
n:nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
 simpl ; apply Rmult_gt_0_lt_compat ; intuition.x, y:R
y_pos:y > 0
x_pos:x > 0
H:0 <= x
H0:x < y
x * 1 > 0
x, y:R
x_encad:0 <= x < y
y_pos:y > 0
x_pos:x = 0
x ^ (2 * 1) < y ^ (2 * 1)
x, y:R
n:nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
x ^ (2 * S (S n)) < y ^ (2 * S (S n)) lra.x, y:R
x_encad:0 <= x < y
y_pos:y > 0
x_pos:x = 0
x ^ (2 * 1) < y ^ (2 * 1)
x, y:R
n:nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
 rewrite x_pos ; rewrite pow_i.x, y:R
x_encad:0 <= x < y
y_pos:y > 0
x_pos:x = 0
0 < y ^ (2 * 1)
x, y:R
x_encad:0 <= x < y
y_pos:y > 0
x_pos:x = 0
(0 < 2 * 1)%nat
x, y:R
n:nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
x ^ (2 * S (S n)) < y ^ (2 * S (S n)) replace (y ^ (2*1)) with (y*y).x, y:R
x_encad:0 <= x < y
y_pos:y > 0
x_pos:x = 0
0 < y * y
x, y:R
x_encad:0 <= x < y
y_pos:y > 0
x_pos:x = 0
y * y = y ^ (2 * 1)
x, y:R
x_encad:0 <= x < y
y_pos:y > 0
x_pos:x = 0
(0 < 2 * 1)%nat
x, y:R
n:nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
 apply Rmult_gt_0_compat ; assumption.x, y:R
x_encad:0 <= x < y
y_pos:y > 0
x_pos:x = 0
y * y = y ^ (2 * 1)
x, y:R
x_encad:0 <= x < y
y_pos:y > 0
x_pos:x = 0
(0 < 2 * 1)%nat
x, y:R
n:nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
 simpl ; field.x, y:R
x_encad:0 <= x < y
y_pos:y > 0
x_pos:x = 0
(0 < 2 * 1)%nat
x, y:R
n:nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
 intuition.x, y:R
n:nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
 assert (Hrew : forall a, a^(2 * S (S n)) = (a ^ 2) * (a ^ (2 * S n))).x, y:R
n:nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
forall a : R,
a ^ (2 * S (S n)) = a ^ 2 * a ^ (2 * S n)
x, y:R
n:nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
Hrew:forall a : R,
a ^ (2 * S (S n)) = a ^ 2 * a ^ (2 * S n)
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
  clear ; intro a ; replace (2 * S (S n))%nat with (S (S (2 * S n)))%nat by intuition.n:nat
a:R
a ^ S (S (2 * S n)) = a ^ 2 * a ^ (2 * S n)
x, y:R
n:nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
Hrew:forall a : R,
a ^ (2 * S (S n)) = a ^ 2 * a ^ (2 * S n)
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
  simpl ; field.x, y:R
n:nat
x_encad:0 <= x < y
x_pos:x >= 0
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
Hrew:forall a : R,
a ^ (2 * S (S n)) = a ^ 2 * a ^ (2 * S n)
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
 case x_pos ; clear x_pos ; intro x_pos.x, y:R
n:nat
x_encad:0 <= x < y
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
Hrew:forall a : R,
a ^ (2 * S (S n)) = a ^ 2 * a ^ (2 * S n)
x_pos:x > 0
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
x, y:R
n:nat
x_encad:0 <= x < y
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
Hrew:forall a : R,
a ^ (2 * S (S n)) = a ^ 2 * a ^ (2 * S n)
x_pos:x = 0
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
 rewrite Hrew ; rewrite Hrew.x, y:R
n:nat
x_encad:0 <= x < y
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
Hrew:forall a : R,
a ^ (2 * S (S n)) = a ^ 2 * a ^ (2 * S n)
x_pos:x > 0
x ^ 2 * x ^ (2 * S n) < y ^ 2 * y ^ (2 * S n)
x, y:R
n:nat
x_encad:0 <= x < y
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
Hrew:forall a : R,
a ^ (2 * S (S n)) = a ^ 2 * a ^ (2 * S n)
x_pos:x = 0
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
 apply Rmult_gt_0_lt_compat ; intuition.x, y:R
n:nat
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
Hrew:forall a : R,
a ^ (2 * S (S n)) = a ^ 2 * a ^ (2 * S n)
x_pos:x > 0
H:0 <= x
H0:x < y
x ^ 2 < y ^ 2
x, y:R
n:nat
x_encad:0 <= x < y
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
Hrew:forall a : R,
a ^ (2 * S (S n)) = a ^ 2 * a ^ (2 * S n)
x_pos:x = 0
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
 apply Rmult_gt_0_lt_compat ; intuition ; lra.x, y:R
n:nat
x_encad:0 <= x < y
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
Hrew:forall a : R,
a ^ (2 * S (S n)) = a ^ 2 * a ^ (2 * S n)
x_pos:x = 0
x ^ (2 * S (S n)) < y ^ (2 * S (S n))
 rewrite x_pos.x, y:R
n:nat
x_encad:0 <= x < y
y_pos:y > 0
IHn:Datan_seq x (S n) < Datan_seq y (S n)
Hrew:forall a : R,
a ^ (2 * S (S n)) = a ^ 2 * a ^ (2 * S n)
x_pos:x = 0
0 ^ (2 * S (S n)) < y ^ (2 * S (S n))
 rewrite pow_i ; intuition.
Qed.

Lemma Datan_seq_decreasing : forall x,  -1 < x -> x < 1 -> Un_decreasing (Datan_seq x).forall x : R,
-1 < x -> x < 1 -> Un_decreasing (Datan_seq x)
Proof.forall x : R,
-1 < x -> x < 1 -> Un_decreasing (Datan_seq x)
intros x x_lb x_ub n.x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
Datan_seq x (S n) <= Datan_seq x n
unfold Datan_seq.x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
x ^ (2 * S n) <= x ^ (2 * n)
replace (2 * S n)%nat with (2 + 2 * n)%nat by ring.x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
x ^ (2 + 2 * n) <= x ^ (2 * n)
rewrite <- (Rmult_1_l (x ^ (2 * n))).x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
x ^ (2 + 2 * n) <= 1 * x ^ (2 * n)
rewrite pow_add.x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
x ^ 2 * x ^ (2 * n) <= 1 * x ^ (2 * n)
apply Rmult_le_compat_r.x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
0 <= x ^ (2 * n)
x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
x ^ 2 <= 1
rewrite pow_mult; apply pow_le, pow2_ge_0.x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
x ^ 2 <= 1
apply Rlt_le; rewrite <- pow2_abs.x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
Rabs x ^ 2 < 1
assert (intabs : 0 <= Rabs x < 1).x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
0 <= Rabs x < 1
x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
intabs:0 <= Rabs x < 1
Rabs x ^ 2 < 1
 split;[apply Rabs_pos | apply Rabs_def1]; tauto.x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
intabs:0 <= Rabs x < 1
Rabs x ^ 2 < 1
apply (pow_lt_1_compat (Rabs x) 2) in intabs.x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
intabs:0 <= Rabs x ^ 2 < 1
Rabs x ^ 2 < 1
x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
intabs:0 <= Rabs x < 1
(0 < 2)%nat
 tauto.x:R
x_lb:-1 < x
x_ub:x < 1
n:nat
intabs:0 <= Rabs x < 1
(0 < 2)%nat
omega.
Qed.

Lemma Datan_seq_CV_0 : forall x, -1 < x -> x < 1 -> Un_cv (Datan_seq x) 0.forall x : R, -1 < x -> x < 1 -> Un_cv (Datan_seq x) 0
Proof.forall x : R, -1 < x -> x < 1 -> Un_cv (Datan_seq x) 0
intros x x_lb x_ub eps eps_pos.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Datan_seq x n) 0 < eps
assert (x_ub2 : Rabs (x^2) < 1).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Rabs (x ^ 2) < 1
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
x_ub2:Rabs (x ^ 2) < 1
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Datan_seq x n) 0 < eps
 rewrite Rabs_pos_eq;[ | apply pow2_ge_0].x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
x ^ 2 < 1
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
x_ub2:Rabs (x ^ 2) < 1
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Datan_seq x n) 0 < eps
 rewrite <- pow2_abs.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Rabs x ^ 2 < 1
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
x_ub2:Rabs (x ^ 2) < 1
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Datan_seq x n) 0 < eps
 assert (H: 0 <= Rabs x < 1)
   by (split;[apply Rabs_pos | apply Rabs_def1; auto]).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
H:0 <= Rabs x < 1
Rabs x ^ 2 < 1
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
x_ub2:Rabs (x ^ 2) < 1
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Datan_seq x n) 0 < eps
 apply (pow_lt_1_compat _ 2) in H;[tauto | omega].x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
x_ub2:Rabs (x ^ 2) < 1
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Datan_seq x n) 0 < eps
elim (pow_lt_1_zero (x^2) x_ub2 eps eps_pos) ; intros N HN ; exists N ; intros n Hn.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
x_ub2:Rabs (x ^ 2) < 1
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat -> Rabs ((x ^ 2) ^ n0) < eps
n:nat
Hn:(n >= N)%nat
R_dist (Datan_seq x n) 0 < eps
unfold R_dist, Datan_seq.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
x_ub2:Rabs (x ^ 2) < 1
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat -> Rabs ((x ^ 2) ^ n0) < eps
n:nat
Hn:(n >= N)%nat
Rabs (x ^ (2 * n) - 0) < eps
replace (x ^ (2 * n) - 0) with ((x ^ 2) ^ n).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
x_ub2:Rabs (x ^ 2) < 1
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat -> Rabs ((x ^ 2) ^ n0) < eps
n:nat
Hn:(n >= N)%nat
Rabs ((x ^ 2) ^ n) < eps
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
x_ub2:Rabs (x ^ 2) < 1
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat -> Rabs ((x ^ 2) ^ n0) < eps
n:nat
Hn:(n >= N)%nat
(x ^ 2) ^ n = x ^ (2 * n) - 0 apply HN ; assumption.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
x_ub2:Rabs (x ^ 2) < 1
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat -> Rabs ((x ^ 2) ^ n0) < eps
n:nat
Hn:(n >= N)%nat
(x ^ 2) ^ n = x ^ (2 * n) - 0
rewrite pow_mult ; field.
Qed.

Lemma Datan_lim : forall x, -1 < x -> x < 1 ->
    Un_cv (fun N : nat => sum_f_R0 (tg_alt (Datan_seq x)) N) (/ (1 + x ^ 2)).forall x : R,
-1 < x ->
x < 1 ->
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Datan_seq x)) N)
  (/ (1 + x ^ 2))
Proof.forall x : R,
-1 < x ->
x < 1 ->
Un_cv
  (fun N : nat => sum_f_R0 (tg_alt (Datan_seq x)) N)
  (/ (1 + x ^ 2))
intros x x_lb x_ub eps eps_pos.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
    (/ (1 + x ^ 2)) < eps
assert (Tool0 : 0 <= x ^ 2) by apply pow2_ge_0.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
    (/ (1 + x ^ 2)) < eps
assert (Tool1 : 0 < (1 + x ^ 2)).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
0 < 1 + x ^ 2
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
    (/ (1 + x ^ 2)) < eps
 solve[apply Rplus_lt_le_0_compat ; intuition].x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
    (/ (1 + x ^ 2)) < eps
assert (Tool2 : / (1 + x ^ 2) > 0).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
/ (1 + x ^ 2) > 0
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
    (/ (1 + x ^ 2)) < eps
 apply Rinv_0_lt_compat ; tauto.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
    (/ (1 + x ^ 2)) < eps
assert (x_ub2' : 0<= Rabs (x^2) < 1).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
0 <= Rabs (x ^ 2) < 1
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
    (/ (1 + x ^ 2)) < eps
 rewrite Rabs_pos_eq, <- pow2_abs;[ | apply pow2_ge_0].x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
0 <= Rabs x ^ 2 < 1
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
    (/ (1 + x ^ 2)) < eps
 apply pow_lt_1_compat;[split;[apply Rabs_pos | ] | omega].x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
Rabs x < 1
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
    (/ (1 + x ^ 2)) < eps
 apply Rabs_def1; assumption.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
    (/ (1 + x ^ 2)) < eps
assert (x_ub2 : Rabs (x^2) < 1) by tauto.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
    (/ (1 + x ^ 2)) < eps
assert (eps'_pos : ((1+x^2)*eps) > 0).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
(1 + x ^ 2) * eps > 0
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
    (/ (1 + x ^ 2)) < eps
  apply Rmult_gt_0_compat ; assumption.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
    (/ (1 + x ^ 2)) < eps
elim (pow_lt_1_zero _ x_ub2 _ eps'_pos) ; intros N HN ; exists N.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n : nat,
(n >= N)%nat ->
Rabs ((x ^ 2) ^ n) < (1 + x ^ 2) * eps
forall n : nat,
(n >= N)%nat ->
R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
  (/ (1 + x ^ 2)) < eps
intros n Hn.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
  (/ (1 + x ^ 2)) < eps
assert (H1 : - x^2 <> 1).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
- x ^ 2 <> 1
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
  (/ (1 + x ^ 2)) < eps
 apply Rlt_not_eq; apply Rle_lt_trans with (2 := Rlt_0_1).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
- x ^ 2 <= 0
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
  (/ (1 + x ^ 2)) < eps
assert (t := pow2_ge_0 x); lra.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
R_dist (sum_f_R0 (tg_alt (Datan_seq x)) n)
  (/ (1 + x ^ 2)) < eps
rewrite Datan_sum_eq.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
R_dist ((1 - (- x ^ 2) ^ S n) / (1 + x ^ 2))
  (/ (1 + x ^ 2)) < eps 
unfold R_dist.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
Rabs
  ((1 - (- x ^ 2) ^ S n) / (1 + x ^ 2) - / (1 + x ^ 2)) <
eps
assert (tool : forall a b, a / b - /b = (-1 + a) /b).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
forall a b : R, a / b - / b = (-1 + a) / b
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b : R, a / b - / b = (-1 + a) / b
Rabs
  ((1 - (- x ^ 2) ^ S n) / (1 + x ^ 2) - / (1 + x ^ 2)) <
eps
 intros a b; rewrite <- (Rmult_1_l (/b)); unfold Rdiv, Rminus.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
a, b:R
a * / b + - (1 * / b) = (-1 + a) * / b
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b : R, a / b - / b = (-1 + a) / b
Rabs
  ((1 - (- x ^ 2) ^ S n) / (1 + x ^ 2) - / (1 + x ^ 2)) <
eps
 rewrite <- Ropp_mult_distr_l_reverse, Rmult_plus_distr_r, Rplus_comm.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
a, b:R
- (1) * / b + a * / b = -1 * / b + a * / b
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b : R, a / b - / b = (-1 + a) / b
Rabs
  ((1 - (- x ^ 2) ^ S n) / (1 + x ^ 2) - / (1 + x ^ 2)) <
eps
 reflexivity.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b : R, a / b - / b = (-1 + a) / b
Rabs
  ((1 - (- x ^ 2) ^ S n) / (1 + x ^ 2) - / (1 + x ^ 2)) <
eps
set (u := 1 + x ^ 2); rewrite tool; unfold Rminus; rewrite <- Rplus_assoc.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b : R, a / b - / b = (-1 + a) / b
u:=1 + x ^ 2:R
Rabs ((-1 + 1 + - (- x ^ 2) ^ S n) / u) < eps
unfold Rdiv, u.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b : R, a / b - / b = (-1 + a) / b
u:=1 + x ^ 2:R
Rabs ((-1 + 1 + - (- x ^ 2) ^ S n) * / (1 + x ^ 2)) <
eps
change (-1) with (-(1)).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b : R, a / b - / b = (-1 + a) / b
u:=1 + x ^ 2:R
Rabs ((- (1) + 1 + - (- x ^ 2) ^ S n) * / (1 + x ^ 2)) <
eps
rewrite Rplus_opp_l, Rplus_0_l, Ropp_mult_distr_l_reverse, Rabs_Ropp.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b : R, a / b - / b = (-1 + a) / b
u:=1 + x ^ 2:R
Rabs ((- x ^ 2) ^ S n * / (1 + x ^ 2)) < eps
rewrite Rabs_mult; clear tool u.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
Rabs ((- x ^ 2) ^ S n) * Rabs (/ (1 + x ^ 2)) < eps
assert (tool : forall k, Rabs ((-x ^ 2) ^ k) = Rabs ((x ^ 2) ^ k)).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
forall k : nat,
Rabs ((- x ^ 2) ^ k) = Rabs ((x ^ 2) ^ k)
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall k : nat,
Rabs ((- x ^ 2) ^ k) = Rabs ((x ^ 2) ^ k)
Rabs ((- x ^ 2) ^ S n) * Rabs (/ (1 + x ^ 2)) < eps
 clear -Tool0; induction k;[simpl; rewrite Rabs_R1;tauto | ].x:R
Tool0:0 <= x ^ 2
k:nat
IHk:Rabs ((- x ^ 2) ^ k) = Rabs ((x ^ 2) ^ k)
Rabs ((- x ^ 2) ^ S k) = Rabs ((x ^ 2) ^ S k)
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall k : nat,
Rabs ((- x ^ 2) ^ k) = Rabs ((x ^ 2) ^ k)
Rabs ((- x ^ 2) ^ S n) * Rabs (/ (1 + x ^ 2)) < eps
 rewrite <- !(tech_pow_Rmult _ k), !Rabs_mult, Rabs_Ropp, IHk, Rabs_pos_eq.x:R
Tool0:0 <= x ^ 2
k:nat
IHk:Rabs ((- x ^ 2) ^ k) = Rabs ((x ^ 2) ^ k)
x ^ 2 * Rabs ((x ^ 2) ^ k) =
x ^ 2 * Rabs ((x ^ 2) ^ k)
x:R
Tool0:0 <= x ^ 2
k:nat
IHk:Rabs ((- x ^ 2) ^ k) = Rabs ((x ^ 2) ^ k)
0 <= x ^ 2
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall k : nat,
Rabs ((- x ^ 2) ^ k) = Rabs ((x ^ 2) ^ k)
Rabs ((- x ^ 2) ^ S n) * Rabs (/ (1 + x ^ 2)) < eps
  reflexivity.x:R
Tool0:0 <= x ^ 2
k:nat
IHk:Rabs ((- x ^ 2) ^ k) = Rabs ((x ^ 2) ^ k)
0 <= x ^ 2
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall k : nat,
Rabs ((- x ^ 2) ^ k) = Rabs ((x ^ 2) ^ k)
Rabs ((- x ^ 2) ^ S n) * Rabs (/ (1 + x ^ 2)) < eps
 exact Tool0.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall k : nat,
Rabs ((- x ^ 2) ^ k) = Rabs ((x ^ 2) ^ k)
Rabs ((- x ^ 2) ^ S n) * Rabs (/ (1 + x ^ 2)) < eps
rewrite tool, (Rabs_pos_eq (/ _)); clear tool;[ | apply Rlt_le; assumption].x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
Rabs ((x ^ 2) ^ S n) * / (1 + x ^ 2) < eps
assert (tool : forall a b c, 0 < b -> a < b * c -> a * / b < c).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
forall a b c : R, 0 < b -> a < b * c -> a * / b < c
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b c : R,
0 < b -> a < b * c -> a * / b < c
Rabs ((x ^ 2) ^ S n) * / (1 + x ^ 2) < eps
 intros a b c bp h; replace c with (b * c * /b).x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
a, b, c:R
bp:0 < b
h:a < b * c
a * / b < b * c * / b
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
a, b, c:R
bp:0 < b
h:a < b * c
b * c * / b = c
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b c : R,
0 < b -> a < b * c -> a * / b < c
Rabs ((x ^ 2) ^ S n) * / (1 + x ^ 2) < eps
  apply Rmult_lt_compat_r.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
a, b, c:R
bp:0 < b
h:a < b * c
0 < / b
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
a, b, c:R
bp:0 < b
h:a < b * c
a < b * c
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
a, b, c:R
bp:0 < b
h:a < b * c
b * c * / b = c
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b c : R,
0 < b -> a < b * c -> a * / b < c
Rabs ((x ^ 2) ^ S n) * / (1 + x ^ 2) < eps  
   apply Rinv_0_lt_compat; assumption.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
a, b, c:R
bp:0 < b
h:a < b * c
a < b * c
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
a, b, c:R
bp:0 < b
h:a < b * c
b * c * / b = c
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b c : R,
0 < b -> a < b * c -> a * / b < c
Rabs ((x ^ 2) ^ S n) * / (1 + x ^ 2) < eps
  assumption.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
a, b, c:R
bp:0 < b
h:a < b * c
b * c * / b = c
x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b c : R,
0 < b -> a < b * c -> a * / b < c
Rabs ((x ^ 2) ^ S n) * / (1 + x ^ 2) < eps
 field; apply Rgt_not_eq; exact bp.x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b c : R,
0 < b -> a < b * c -> a * / b < c
Rabs ((x ^ 2) ^ S n) * / (1 + x ^ 2) < eps
apply tool;[exact Tool1 | ].x:R
x_lb:-1 < x
x_ub:x < 1
eps:R
eps_pos:eps > 0
Tool0:0 <= x ^ 2
Tool1:0 < 1 + x ^ 2
Tool2:/ (1 + x ^ 2) > 0
x_ub2':0 <= Rabs (x ^ 2) < 1
x_ub2:Rabs (x ^ 2) < 1
eps'_pos:(1 + x ^ 2) * eps > 0
N:nat
HN:forall n0 : nat,
(n0 >= N)%nat ->
Rabs ((x ^ 2) ^ n0) < (1 + x ^ 2) * eps
n:nat
Hn:(n >= N)%nat
H1:- x ^ 2 <> 1
tool:forall a b c : R,
0 < b -> a < b * c -> a * / b < c
Rabs ((x ^ 2) ^ S n) < (1 + x ^ 2) * eps
apply HN; omega.
Qed.

Lemma Datan_CVU_prelim : forall c (r : posreal), Rabs c + r < 1 ->
 CVU (fun N x => sum_f_R0 (tg_alt (Datan_seq x)) N)
     (fun y : R => / (1 + y ^ 2)) c r.forall (c : R) (r : posreal),
Rabs c + r < 1 ->
CVU
  (fun (N : nat) (x : R) =>
   sum_f_R0 (tg_alt (Datan_seq x)) N)
  (fun y : R => / (1 + y ^ 2)) c r
Proof.forall (c : R) (r : posreal),
Rabs c + r < 1 ->
CVU
  (fun (N : nat) (x : R) =>
   sum_f_R0 (tg_alt (Datan_seq x)) N)
  (fun y : R => / (1 + y ^ 2)) c r
intros c r ub_ub eps eps_pos.c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
exists N : nat,
  forall (n : nat) (y : R),
  (N <= n)%nat ->
  Boule c r y ->
  Rabs
    (/ (1 + y ^ 2) - sum_f_R0 (tg_alt (Datan_seq y)) n) <
  eps
apply (Alt_CVU (fun x n => Datan_seq n x) 
         (fun x => /(1 + x ^ 2))
         (Datan_seq (Rabs c + r)) c r).c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
forall x : R,
Boule c r x ->
Un_decreasing (fun n : nat => Datan_seq x n)
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
forall x : R,
Boule c r x -> Un_cv (fun n : nat => Datan_seq x n) 0
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
forall x : R,
Boule c r x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => Datan_seq x i)))
  (/ (1 + x ^ 2))
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
forall (x : R) (n : nat),
Boule c r x ->
Datan_seq x n <= Datan_seq (Rabs c + r) n
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Un_cv (Datan_seq (Rabs c + r)) 0
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
     intros x inb; apply Datan_seq_decreasing;
      try (apply Boule_lt in inb; apply Rabs_def2 in inb;
         destruct inb; lra).c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
forall x : R,
Boule c r x -> Un_cv (fun n : nat => Datan_seq x n) 0
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
forall x : R,
Boule c r x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => Datan_seq x i)))
  (/ (1 + x ^ 2))
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
forall (x : R) (n : nat),
Boule c r x ->
Datan_seq x n <= Datan_seq (Rabs c + r) n
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Un_cv (Datan_seq (Rabs c + r)) 0
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
    intros x inb; apply Datan_seq_CV_0;
      try (apply Boule_lt in inb; apply Rabs_def2 in inb;
         destruct inb; lra).c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
forall x : R,
Boule c r x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => Datan_seq x i)))
  (/ (1 + x ^ 2))
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
forall (x : R) (n : nat),
Boule c r x ->
Datan_seq x n <= Datan_seq (Rabs c + r) n
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Un_cv (Datan_seq (Rabs c + r)) 0
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
   intros x inb; apply (Datan_lim x);
      try (apply Boule_lt in inb; apply Rabs_def2 in inb;
         destruct inb; lra).c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
forall (x : R) (n : nat),
Boule c r x ->
Datan_seq x n <= Datan_seq (Rabs c + r) n
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Un_cv (Datan_seq (Rabs c + r)) 0
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
  intros x [ | n] inb.c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
x:R
inb:Boule c r x
Datan_seq x 0 <= Datan_seq (Rabs c + r) 0
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
x:R
n:nat
inb:Boule c r x
Datan_seq x (S n) <= Datan_seq (Rabs c + r) (S n)
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Un_cv (Datan_seq (Rabs c + r)) 0
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
   solve[unfold Datan_seq; apply Rle_refl].c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
x:R
n:nat
inb:Boule c r x
Datan_seq x (S n) <= Datan_seq (Rabs c + r) (S n)
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Un_cv (Datan_seq (Rabs c + r)) 0
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
  rewrite <- (Datan_seq_Rabs x); apply Rlt_le, Datan_seq_increasing.c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
x:R
n:nat
inb:Boule c r x
(S n > 0)%nat
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
x:R
n:nat
inb:Boule c r x
0 <= Rabs x < Rabs c + r
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Un_cv (Datan_seq (Rabs c + r)) 0
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
   omega.c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
x:R
n:nat
inb:Boule c r x
0 <= Rabs x < Rabs c + r
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Un_cv (Datan_seq (Rabs c + r)) 0
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
  apply Boule_lt in inb; intuition.c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
x:R
n:nat
inb:Rabs x < Rabs c + r
0 <= Rabs x
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Un_cv (Datan_seq (Rabs c + r)) 0
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
  solve[apply Rabs_pos].c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Un_cv (Datan_seq (Rabs c + r)) 0
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
 apply Datan_seq_CV_0.c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
-1 < Rabs c + r
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Rabs c + r < 1
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
  apply Rlt_trans with 0;[lra | ].c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < Rabs c + r
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Rabs c + r < 1
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
  apply Rplus_le_lt_0_compat.c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 <= Rabs c
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < r
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Rabs c + r < 1
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
   solve[apply Rabs_pos].c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < r
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Rabs c + r < 1
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
  destruct r; assumption.c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
Rabs c + r < 1
c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
 assumption.c:R
r:posreal
ub_ub:Rabs c + r < 1
eps:R
eps_pos:0 < eps
0 < eps
assumption.
Qed.

Lemma Datan_is_datan : forall (N:nat) (x:R),
      -1 <= x ->
      x < 1 ->
derivable_pt_lim (fun x => sum_f_R0 (tg_alt (Ratan_seq x)) N) x (sum_f_R0 (tg_alt (Datan_seq x)) N).forall (N : nat) (x : R),
-1 <= x ->
x < 1 ->
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
Proof.forall (N : nat) (x : R),
-1 <= x ->
x < 1 ->
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
assert (Tool : forall N, (-1) ^ (S (2 * N))  = - 1).forall N : nat, (-1) ^ S (2 * N) = -1
Tool:forall N : nat, (-1) ^ S (2 * N) = -1
forall (N : nat) (x : R),
-1 <= x ->
x < 1 ->
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
 intro n ; induction n.(-1) ^ S (2 * 0) = -1
n:nat
IHn:(-1) ^ S (2 * n) = -1
(-1) ^ S (2 * S n) = -1
Tool:forall N : nat, (-1) ^ S (2 * N) = -1
forall (N : nat) (x : R),
-1 <= x ->
x < 1 ->
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
  simpl ; field.n:nat
IHn:(-1) ^ S (2 * n) = -1
(-1) ^ S (2 * S n) = -1
Tool:forall N : nat, (-1) ^ S (2 * N) = -1
forall (N : nat) (x : R),
-1 <= x ->
x < 1 ->
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
  replace ((-1) ^ S (2 * S n)) with ((-1) ^ 2 * (-1) ^ S (2*n)).n:nat
IHn:(-1) ^ S (2 * n) = -1
(-1) ^ 2 * (-1) ^ S (2 * n) = -1
n:nat
IHn:(-1) ^ S (2 * n) = -1
(-1) ^ 2 * (-1) ^ S (2 * n) = (-1) ^ S (2 * S n)
Tool:forall N : nat, (-1) ^ S (2 * N) = -1
forall (N : nat) (x : R),
-1 <= x ->
x < 1 ->
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
  rewrite IHn ; field.n:nat
IHn:(-1) ^ S (2 * n) = -1
(-1) ^ 2 * (-1) ^ S (2 * n) = (-1) ^ S (2 * S n)
Tool:forall N : nat, (-1) ^ S (2 * N) = -1
forall (N : nat) (x : R),
-1 <= x ->
x < 1 ->
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
  rewrite <- pow_add.n:nat
IHn:(-1) ^ S (2 * n) = -1
(-1) ^ (2 + S (2 * n)) = (-1) ^ S (2 * S n)
Tool:forall N : nat, (-1) ^ S (2 * N) = -1
forall (N : nat) (x : R),
-1 <= x ->
x < 1 ->
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
  replace (2 + S (2 * n))%nat with (S (2 * S n))%nat.n:nat
IHn:(-1) ^ S (2 * n) = -1
(-1) ^ S (2 * S n) = (-1) ^ S (2 * S n)
n:nat
IHn:(-1) ^ S (2 * n) = -1
S (2 * S n) = (2 + S (2 * n))%nat
Tool:forall N : nat, (-1) ^ S (2 * N) = -1
forall (N : nat) (x : R),
-1 <= x ->
x < 1 ->
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
  reflexivity.n:nat
IHn:(-1) ^ S (2 * n) = -1
S (2 * S n) = (2 + S (2 * n))%nat
Tool:forall N : nat, (-1) ^ S (2 * N) = -1
forall (N : nat) (x : R),
-1 <= x ->
x < 1 ->
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
  intuition.Tool:forall N : nat, (-1) ^ S (2 * N) = -1
forall (N : nat) (x : R),
-1 <= x ->
x < 1 ->
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
intros N x x_lb x_ub.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
 induction N.Tool:forall N : nat, (-1) ^ S (2 * N) = -1
x:R
x_lb:-1 <= x
x_ub:x < 1
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) 0) x
  (sum_f_R0 (tg_alt (Datan_seq x)) 0)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) (S N)) x
  (sum_f_R0 (tg_alt (Datan_seq x)) (S N))
  unfold Datan_seq, Ratan_seq, tg_alt ; simpl.Tool:forall N : nat, (-1) ^ S (2 * N) = -1
x:R
x_lb:-1 <= x
x_ub:x < 1
derivable_pt_lim (fun x0 : R => 1 * (x0 * 1 / 1)) x
  (1 * 1)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) (S N)) x
  (sum_f_R0 (tg_alt (Datan_seq x)) (S N))
  intros eps eps_pos.Tool:forall N : nat, (-1) ^ S (2 * N) = -1
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((1 * ((x + h) * 1 / 1) - 1 * (x * 1 / 1)) / h -
     1 * 1) < eps
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) (S N)) x
  (sum_f_R0 (tg_alt (Datan_seq x)) (S N))
   elim (derivable_pt_lim_id x eps eps_pos) ; intros delta Hdelta ; exists delta.Tool:forall N : nat, (-1) ^ S (2 * N) = -1
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
delta:posreal
Hdelta:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((id (x + h) - id x) / h - 1) < eps
forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs
  ((1 * ((x + h) * 1 / 1) - 1 * (x * 1 / 1)) / h -
   1 * 1) < eps
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) (S N)) x
  (sum_f_R0 (tg_alt (Datan_seq x)) (S N))
   intros h hneq h_b.Tool:forall N : nat, (-1) ^ S (2 * N) = -1
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((id (x + h0) - id x) / h0 - 1) < eps
h:R
hneq:h <> 0
h_b:Rabs h < delta
Rabs
  ((1 * ((x + h) * 1 / 1) - 1 * (x * 1 / 1)) / h -
   1 * 1) < eps
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) (S N)) x
  (sum_f_R0 (tg_alt (Datan_seq x)) (S N))
    replace (1 * ((x + h) * 1 / 1) - 1 * (x * 1 / 1)) with (id (x + h) - id x).Tool:forall N : nat, (-1) ^ S (2 * N) = -1
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((id (x + h0) - id x) / h0 - 1) < eps
h:R
hneq:h <> 0
h_b:Rabs h < delta
Rabs ((id (x + h) - id x) / h - 1 * 1) < eps
Tool:forall N : nat, (-1) ^ S (2 * N) = -1
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((id (x + h0) - id x) / h0 - 1) < eps
h:R
hneq:h <> 0
h_b:Rabs h < delta
id (x + h) - id x =
1 * ((x + h) * 1 / 1) - 1 * (x * 1 / 1)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) (S N)) x
  (sum_f_R0 (tg_alt (Datan_seq x)) (S N)) 
    rewrite Rmult_1_r.Tool:forall N : nat, (-1) ^ S (2 * N) = -1
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((id (x + h0) - id x) / h0 - 1) < eps
h:R
hneq:h <> 0
h_b:Rabs h < delta
Rabs ((id (x + h) - id x) / h - 1) < eps
Tool:forall N : nat, (-1) ^ S (2 * N) = -1
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((id (x + h0) - id x) / h0 - 1) < eps
h:R
hneq:h <> 0
h_b:Rabs h < delta
id (x + h) - id x =
1 * ((x + h) * 1 / 1) - 1 * (x * 1 / 1)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) (S N)) x
  (sum_f_R0 (tg_alt (Datan_seq x)) (S N))
    apply Hdelta ; assumption.Tool:forall N : nat, (-1) ^ S (2 * N) = -1
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs ((id (x + h0) - id x) / h0 - 1) < eps
h:R
hneq:h <> 0
h_b:Rabs h < delta
id (x + h) - id x =
1 * ((x + h) * 1 / 1) - 1 * (x * 1 / 1)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) (S N)) x
  (sum_f_R0 (tg_alt (Datan_seq x)) (S N))
    unfold id ; field ; assumption.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) (S N)) x
  (sum_f_R0 (tg_alt (Datan_seq x)) (S N))
  intros eps eps_pos.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
  assert (eps_3_pos : (eps/3) > 0) by lra.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
  elim (IHN (eps/3) eps_3_pos) ; intros delta1 Hdelta1.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
  assert (Main : derivable_pt_lim (fun x : R =>tg_alt (Ratan_seq x) (S N)) x ((tg_alt (Datan_seq x)) (S N))).Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   clear -Tool ; intros eps' eps'_pos.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((tg_alt (Ratan_seq (x + h)) (S N) -
      tg_alt (Ratan_seq x) (S N)) / h -
     tg_alt (Datan_seq x) (S N)) < eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   elim (derivable_pt_lim_pow x (2 * (S N) + 1) eps' eps'_pos) ; intros delta Hdelta ; exists delta.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs
  (((x + h) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs
  ((tg_alt (Ratan_seq (x + h)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h -
   tg_alt (Datan_seq x) (S N)) < eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   intros h h_neq h_b ; unfold tg_alt, Ratan_seq, Datan_seq.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Rabs
  (((-1) ^ S N *
    ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
    (-1) ^ S N *
    (x ^ (2 * S N + 1) / INR (2 * S N + 1))) / h -
   (-1) ^ S N * x ^ (2 * S N)) < eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   replace (((-1) ^ S N * ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
    (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) / h -
    (-1) ^ S N * x ^ (2 * S N)) 
    with (((-1)^(S N)) * ((((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
    (x ^ (2 * S N + 1) / INR (2 * S N + 1))) / h - x ^ (2 * S N))).Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Rabs
  ((-1) ^ S N *
   (((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
     x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
    x ^ (2 * S N))) < eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   rewrite Rabs_mult ; rewrite pow_1_abs ; rewrite Rmult_1_l.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Rabs
  (((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
    x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
   x ^ (2 * S N)) < eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   replace (((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
    x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h - x ^ (2 * S N))
    with ((/INR (2* S N + 1)) * (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
    INR (2 * S N + 1) * x ^ pred (2 * S N + 1))).Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Rabs
  (/ INR (2 * S N + 1) *
   (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
    INR (2 * S N + 1) *
    x ^ Init.Nat.pred (2 * S N + 1))) < eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
 INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   rewrite Rabs_mult.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Rabs (/ INR (2 * S N + 1)) *
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) <
eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
 INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   case (Req_dec (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
     INR (2 * S N + 1) * x ^ pred (2 * S N + 1)) 0) ; intro Heq.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) = 0
Rabs (/ INR (2 * S N + 1)) *
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) <
eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
Rabs (/ INR (2 * S N + 1)) *
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) <
eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
 INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   rewrite Heq ; rewrite Rabs_R0 ; rewrite Rmult_0_r ; assumption.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
Rabs (/ INR (2 * S N + 1)) *
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) <
eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
 INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   apply Rlt_trans with (r2:=Rabs
           (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
            INR (2 * S N + 1) * x ^ pred (2 * S N + 1))).Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
Rabs (/ INR (2 * S N + 1)) *
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) <
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1))
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) <
eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
 INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   rewrite <- Rmult_1_l ; apply Rmult_lt_compat_r.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
0 <
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1))
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
Rabs (/ INR (2 * S N + 1)) < 1
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) <
eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
 INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   apply Rabs_pos_lt ; assumption.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
Rabs (/ INR (2 * S N + 1)) < 1
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) <
eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
 INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   rewrite Rabs_right.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
/ INR (2 * S N + 1) < 1
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
/ INR (2 * S N + 1) >= 0
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) <
eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
 INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   replace 1 with (/1) by field.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
/ INR (2 * S N + 1) < / 1
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
/ INR (2 * S N + 1) >= 0
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) <
eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
 INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   apply Rinv_1_lt_contravar ; intuition.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
/ INR (2 * S N + 1) >= 0
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) <
eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
 INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   apply Rgt_ge ; replace (INR (2 * S N + 1)) with (INR (2*S N) + 1) ;
   [apply RiemannInt.RinvN_pos | ].Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
INR (2 * S N) + 1 = INR (2 * S N + 1)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) <
eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
 INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   replace (2 * S N + 1)%nat with (S (2 * S N))%nat by intuition ;
   rewrite S_INR ; reflexivity.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Heq:((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) /
h -
INR (2 * S N + 1) *
x ^ Init.Nat.pred (2 * S N + 1) <> 0
Rabs
  (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
   INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) <
eps'
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
 INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   apply Hdelta ; assumption.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
 INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   rewrite Rmult_minus_distr_l.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h) -
/ INR (2 * S N + 1) *
(INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1)) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   replace (/ INR (2 * S N + 1) * (INR (2 * S N + 1) * x ^ pred (2 * S N + 1))) with (x ^ (2 * S N)).Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h) -
x ^ (2 * S N) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
 x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
x ^ (2 * S N) =
/ INR (2 * S N + 1) *
(INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1))
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   unfold Rminus ; rewrite Rplus_comm.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
- x ^ (2 * S N) +
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) + - x ^ (2 * S N + 1)) / h) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) +
 - (x ^ (2 * S N + 1) / INR (2 * S N + 1))) / h +
- x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
x ^ (2 * S N) =
/ INR (2 * S N + 1) *
(INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1))
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   replace (((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) +
      - (x ^ (2 * S N + 1) / INR (2 * S N + 1))) / h + - x ^ (2 * S N))
      with (- x ^ (2 * S N) + (((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) +
      - (x ^ (2 * S N + 1) / INR (2 * S N + 1))) / h)) by intuition.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
- x ^ (2 * S N) +
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) + - x ^ (2 * S N + 1)) / h) =
- x ^ (2 * S N) +
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) +
 - (x ^ (2 * S N + 1) / INR (2 * S N + 1))) / h
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
x ^ (2 * S N) =
/ INR (2 * S N + 1) *
(INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1))
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   apply Rplus_eq_compat_l.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
/ INR (2 * S N + 1) *
(((x + h) ^ (2 * S N + 1) + - x ^ (2 * S N + 1)) / h) =
((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) +
 - (x ^ (2 * S N + 1) / INR (2 * S N + 1))) / h
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
x ^ (2 * S N) =
/ INR (2 * S N + 1) *
(INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1))
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps field.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
INR (2 * S N + 1) <> 0 /\ h <> 0
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
x ^ (2 * S N) =
/ INR (2 * S N + 1) *
(INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1))
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   split ; [apply Rgt_not_eq|] ; intuition.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
x ^ (2 * S N) =
/ INR (2 * S N + 1) *
(INR (2 * S N + 1) * x ^ Init.Nat.pred (2 * S N + 1))
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   clear ; replace (pred (2 * S N + 1)) with (2 * S N)%nat by intuition.N:nat
x:R
x ^ (2 * S N) =
/ INR (2 * S N + 1) *
(INR (2 * S N + 1) * x ^ (2 * S N))
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   field ; apply Rgt_not_eq ; intuition.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x, eps':R
eps'_pos:0 < eps'
delta:posreal
Hdelta:forall h0 : R,
h0 <> 0 ->
Rabs h0 < delta ->
Rabs
  (((x + h0) ^ (2 * S N + 1) -
    x ^ (2 * S N + 1)) / h0 -
   INR (2 * S N + 1) *
   x ^ Init.Nat.pred (2 * S N + 1)) < eps'
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(-1) ^ S N *
(((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
  x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h -
 x ^ (2 * S N)) =
((-1) ^ S N *
 ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
 (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) /
h - (-1) ^ S N * x ^ (2 * S N)
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   field ; split ; [apply Rgt_not_eq |] ; intuition.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   elim (Main (eps/3) eps_3_pos) ; intros delta2 Hdelta2.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:posreal
Hdelta1:forall h : R,
h <> 0 ->
Rabs h < delta1 ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
delta2:posreal
Hdelta2:forall h : R,
h <> 0 ->
Rabs h < delta2 ->
Rabs
  ((tg_alt (Ratan_seq (x + h)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   destruct delta1 as (delta1, delta1_pos) ; destruct delta2 as (delta2, delta2_pos).Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h : R,
h <> 0 ->
Rabs h <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h : R,
h <> 0 ->
Rabs h <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   pose (mydelta := Rmin delta1 delta2).Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h : R,
h <> 0 ->
Rabs h <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h : R,
h <> 0 ->
Rabs h <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   assert (mydelta_pos : mydelta > 0).Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h : R,
h <> 0 ->
Rabs h <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h : R,
h <> 0 ->
Rabs h <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta > 0
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h : R,
h <> 0 ->
Rabs h <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h : R,
h <> 0 ->
Rabs h <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
    unfold mydelta ; rewrite Rmin_Rgt ; split ; assumption.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h : R,
h <> 0 ->
Rabs h <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h : R,
h <> 0 ->
Rabs h <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
      sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
     sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   pose (delta := mkposreal mydelta mydelta_pos) ; exists delta ; intros h h_neq h_b.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
IHN:derivable_pt_lim
  (fun x0 : R =>
   sum_f_R0 (tg_alt (Ratan_seq x0)) N) x
  (sum_f_R0 (tg_alt (Datan_seq x)) N)
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
Main:derivable_pt_lim
  (fun x0 : R => tg_alt (Ratan_seq x0) (S N)) x
  (tg_alt (Datan_seq x) (S N))
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
    sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   clear Main IHN.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
    sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h -
   sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   unfold Rminus at 1.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
    sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
   - sum_f_R0 (tg_alt (Datan_seq x)) (S N)) < eps
   apply Rle_lt_trans with (r2:=eps/3 + eps / 3).Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
    sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
   - sum_f_R0 (tg_alt (Datan_seq x)) (S N)) <=
eps / 3 + eps / 3
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
eps / 3 + eps / 3 < eps
   assert (Temp : (sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
    sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
    - sum_f_R0 (tg_alt (Datan_seq x)) (S N) = ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
              sum_f_R0 (tg_alt (Ratan_seq x)) N) / h) + (-
             sum_f_R0 (tg_alt (Datan_seq x)) N) + ((tg_alt (Ratan_seq (x + h)) (S N) - tg_alt (Ratan_seq x) (S N)) /
             h - tg_alt (Datan_seq x) (S N))).Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
(sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
 sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) (S N) =
(sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
 sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) N +
((tg_alt (Ratan_seq (x + h)) (S N) -
  tg_alt (Ratan_seq x) (S N)) / h -
 tg_alt (Datan_seq x) (S N))
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Temp:(sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
 sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) (S N) =
(sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
 sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) N +
((tg_alt (Ratan_seq (x + h)) (S N) -
  tg_alt (Ratan_seq x) (S N)) / h -
 tg_alt (Datan_seq x) (S N))
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
    sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
   - sum_f_R0 (tg_alt (Datan_seq x)) (S N)) <=
eps / 3 + eps / 3
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
eps / 3 + eps / 3 < eps
    simpl ; field ; intuition.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Temp:(sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
 sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) (S N) =
(sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
 sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) N +
((tg_alt (Ratan_seq (x + h)) (S N) -
  tg_alt (Ratan_seq x) (S N)) / h -
 tg_alt (Datan_seq x) (S N))
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
    sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
   - sum_f_R0 (tg_alt (Datan_seq x)) (S N)) <=
eps / 3 + eps / 3
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
eps / 3 + eps / 3 < eps
   apply Rle_trans with (r2:= Rabs ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
        sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
       - sum_f_R0 (tg_alt (Datan_seq x)) N) +
       Rabs ((tg_alt (Ratan_seq (x + h)) (S N) - tg_alt (Ratan_seq x) (S N)) / h -
        tg_alt (Datan_seq x) (S N))).Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Temp:(sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
 sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) (S N) =
(sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
 sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) N +
((tg_alt (Ratan_seq (x + h)) (S N) -
  tg_alt (Ratan_seq x) (S N)) / h -
 tg_alt (Datan_seq x) (S N))
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
    sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
   - sum_f_R0 (tg_alt (Datan_seq x)) (S N)) <=
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
   - sum_f_R0 (tg_alt (Datan_seq x)) N) +
Rabs
  ((tg_alt (Ratan_seq (x + h)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h -
   tg_alt (Datan_seq x) (S N))
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Temp:(sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
 sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) (S N) =
(sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
 sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) N +
((tg_alt (Ratan_seq (x + h)) (S N) -
  tg_alt (Ratan_seq x) (S N)) / h -
 tg_alt (Datan_seq x) (S N))
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
   - sum_f_R0 (tg_alt (Datan_seq x)) N) +
Rabs
  ((tg_alt (Ratan_seq (x + h)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h -
   tg_alt (Datan_seq x) (S N)) <= 
eps / 3 + eps / 3
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
eps / 3 + eps / 3 < eps
   rewrite Temp ; clear Temp ; apply Rabs_triang.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
Temp:(sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
 sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) (S N) =
(sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
 sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) N +
((tg_alt (Ratan_seq (x + h)) (S N) -
  tg_alt (Ratan_seq x) (S N)) / h -
 tg_alt (Datan_seq x) (S N))
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
   - sum_f_R0 (tg_alt (Datan_seq x)) N) +
Rabs
  ((tg_alt (Ratan_seq (x + h)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h -
   tg_alt (Datan_seq x) (S N)) <= 
eps / 3 + eps / 3
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
eps / 3 + eps / 3 < eps
   apply Rplus_le_compat ; apply Rlt_le ; [apply Hdelta1 | apply Hdelta2] ;
   intuition ; apply Rlt_le_trans with (r2:=delta) ; intuition unfold delta, mydelta.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h = 0 -> False
h_b:Rabs h < delta
Temp:(sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
 sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) (S N) =
(sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
 sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) N +
((tg_alt (Ratan_seq (x + h)) (S N) -
  tg_alt (Ratan_seq x) (S N)) / h -
 tg_alt (Datan_seq x) (S N))
{|
pos := Rmin delta1 delta2;
cond_pos := mydelta_pos |} <=
{| pos := delta1; cond_pos := delta1_pos |}
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h = 0 -> False
h_b:Rabs h < delta
Temp:(sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
 sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) (S N) =
(sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
 sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) N +
((tg_alt (Ratan_seq (x + h)) (S N) -
  tg_alt (Ratan_seq x) (S N)) / h -
 tg_alt (Datan_seq x) (S N))
{|
pos := Rmin delta1 delta2;
cond_pos := mydelta_pos |} <=
{| pos := delta2; cond_pos := delta2_pos |}
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
eps / 3 + eps / 3 < eps
   apply Rmin_l.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
(h0 = 0 -> False) ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h = 0 -> False
h_b:Rabs h < delta
Temp:(sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
 sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) (S N) =
(sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
 sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
- sum_f_R0 (tg_alt (Datan_seq x)) N +
((tg_alt (Ratan_seq (x + h)) (S N) -
  tg_alt (Ratan_seq x) (S N)) / h -
 tg_alt (Datan_seq x) (S N))
{|
pos := Rmin delta1 delta2;
cond_pos := mydelta_pos |} <=
{| pos := delta2; cond_pos := delta2_pos |}
Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
eps / 3 + eps / 3 < eps
   apply Rmin_r.Tool:forall N0 : nat, (-1) ^ S (2 * N0) = -1
N:nat
x:R
x_lb:-1 <= x
x_ub:x < 1
eps:R
eps_pos:0 < eps
eps_3_pos:eps / 3 > 0
delta1:R
delta1_pos:0 < delta1
Hdelta1:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta1; cond_pos := delta1_pos |} ->
Rabs
  ((sum_f_R0 (tg_alt (Ratan_seq (x + h0))) N -
    sum_f_R0 (tg_alt (Ratan_seq x)) N) / h0 -
   sum_f_R0 (tg_alt (Datan_seq x)) N) <
eps / 3
delta2:R
delta2_pos:0 < delta2
Hdelta2:forall h0 : R,
h0 <> 0 ->
Rabs h0 <
{| pos := delta2; cond_pos := delta2_pos |} ->
Rabs
  ((tg_alt (Ratan_seq (x + h0)) (S N) -
    tg_alt (Ratan_seq x) (S N)) / h0 -
   tg_alt (Datan_seq x) (S N)) < 
eps / 3
mydelta:=Rmin delta1 delta2:R
mydelta_pos:mydelta > 0
delta:={| pos := mydelta; cond_pos := mydelta_pos |}:posreal
h:R
h_neq:h <> 0
h_b:Rabs h < delta
eps / 3 + eps / 3 < eps
   lra.
Qed.

Lemma Ratan_CVU' :
  CVU (fun N x => sum_f_R0 (tg_alt (Ratan_seq x)) N)
                     ps_atan (/2) (mkposreal (/2) pos_half_prf).CVU
  (fun (N : nat) (x : R) =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) ps_atan (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |}
Proof.CVU
  (fun (N : nat) (x : R) =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) ps_atan (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |}
apply (Alt_CVU (fun i r => Ratan_seq r i) ps_atan PI_tg (/2) pos_half);
  lazy beta.forall x : R,
Boule (/ 2) pos_half x ->
Un_decreasing (fun n : nat => Ratan_seq x n)
forall x : R,
Boule (/ 2) pos_half x ->
Un_cv (fun n : nat => Ratan_seq x n) 0
forall x : R,
Boule (/ 2) pos_half x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => Ratan_seq x i)))
  (ps_atan x)
forall (x : R) (n : nat),
Boule (/ 2) pos_half x -> Ratan_seq x n <= PI_tg n
Un_cv PI_tg 0
    now intros; apply Ratan_seq_decreasing, Boule_half_to_interval.forall x : R,
Boule (/ 2) pos_half x ->
Un_cv (fun n : nat => Ratan_seq x n) 0
forall x : R,
Boule (/ 2) pos_half x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => Ratan_seq x i)))
  (ps_atan x)
forall (x : R) (n : nat),
Boule (/ 2) pos_half x -> Ratan_seq x n <= PI_tg n
Un_cv PI_tg 0
   now intros; apply Ratan_seq_converging, Boule_half_to_interval.forall x : R,
Boule (/ 2) pos_half x ->
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => Ratan_seq x i)))
  (ps_atan x)
forall (x : R) (n : nat),
Boule (/ 2) pos_half x -> Ratan_seq x n <= PI_tg n
Un_cv PI_tg 0
  intros x b; apply Boule_half_to_interval in b.x:R
b:0 <= x <= 1
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => Ratan_seq x i)))
  (ps_atan x)
forall (x : R) (n : nat),
Boule (/ 2) pos_half x -> Ratan_seq x n <= PI_tg n
Un_cv PI_tg 0
  unfold ps_atan; destruct (in_int x) as [inside | outside];
   [ | destruct b; case outside; split; lra].x:R
b:0 <= x <= 1
inside:-1 <= x <= 1
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => Ratan_seq x i)))
  (let (v, _) := ps_atan_exists_1 x inside in v)
forall (x : R) (n : nat),
Boule (/ 2) pos_half x -> Ratan_seq x n <= PI_tg n
Un_cv PI_tg 0
  destruct (ps_atan_exists_1 x inside) as [v Pv].x:R
b:0 <= x <= 1
inside:-1 <= x <= 1
v:R
Pv:Un_cv
  (fun N : nat =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) v
Un_cv
  (sum_f_R0 (tg_alt (fun i : nat => Ratan_seq x i))) v
forall (x : R) (n : nat),
Boule (/ 2) pos_half x -> Ratan_seq x n <= PI_tg n
Un_cv PI_tg 0
  apply Un_cv_ext with (2 := Pv);[reflexivity].forall (x : R) (n : nat),
Boule (/ 2) pos_half x -> Ratan_seq x n <= PI_tg n
Un_cv PI_tg 0
 intros x n b; apply Boule_half_to_interval in b.x:R
n:nat
b:0 <= x <= 1
Ratan_seq x n <= PI_tg n
Un_cv PI_tg 0
 rewrite <- (Rmult_1_l (PI_tg n)); unfold Ratan_seq, PI_tg.x:R
n:nat
b:0 <= x <= 1
x ^ (2 * n + 1) / INR (2 * n + 1) <=
1 * / INR (2 * n + 1)
Un_cv PI_tg 0 
 apply Rmult_le_compat_r.x:R
n:nat
b:0 <= x <= 1
0 <= / INR (2 * n + 1)
x:R
n:nat
b:0 <= x <= 1
x ^ (2 * n + 1) <= 1
Un_cv PI_tg 0
  apply Rlt_le, Rinv_0_lt_compat, (lt_INR 0); omega.x:R
n:nat
b:0 <= x <= 1
x ^ (2 * n + 1) <= 1
Un_cv PI_tg 0
 rewrite <- (pow1 (2 * n + 1)); apply pow_incr; assumption.Un_cv PI_tg 0
exact PI_tg_cv.
Qed.

Lemma Ratan_CVU :
  CVU (fun N x => sum_f_R0 (tg_alt (Ratan_seq x)) N)
                     ps_atan 0  (mkposreal 1 Rlt_0_1).CVU
  (fun (N : nat) (x : R) =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) ps_atan 0
  {| pos := 1; cond_pos := Rlt_0_1 |}
Proof.CVU
  (fun (N : nat) (x : R) =>
   sum_f_R0 (tg_alt (Ratan_seq x)) N) ps_atan 0
  {| pos := 1; cond_pos := Rlt_0_1 |}
intros eps ep; destruct (Ratan_CVU' eps ep) as [N Pn].eps:R
ep:0 < eps
N:nat
Pn:forall (n : nat) (y : R),
(N <= n)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n) <
eps
exists N0 : nat,
  forall (n : nat) (y : R),
  (N0 <= n)%nat ->
  Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} y ->
  Rabs (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n) <
  eps
exists N; intros n x nN b_y.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
case (Rtotal_order 0 x) as [xgt0 | [x0 | x0]].eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
xgt0:0 < x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 = x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
  assert (Boule (/2) {| pos := / 2; cond_pos := pos_half_prf|} x).eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
xgt0:0 < x
Boule (/ 2) {| pos := / 2; cond_pos := pos_half_prf |}
  x
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
xgt0:0 < x
H:Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 = x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
   revert b_y; unfold Boule; simpl; intros b_y; apply Rabs_def2 in b_y.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
xgt0:0 < x
b_y:x - 0 < 1 /\ - (1) < x - 0
Rabs (x - / 2) < / 2
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
xgt0:0 < x
H:Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 = x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
   destruct b_y; unfold Boule; simpl; apply Rabs_def1; lra.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
xgt0:0 < x
H:Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 = x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
  apply Pn; assumption.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 = x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
 rewrite <- x0, ps_atan0_0.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 = x
Rabs (0 - sum_f_R0 (tg_alt (Ratan_seq 0)) n) < eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
 rewrite <- (sum_eq (fun _ => 0)), sum_cte, Rmult_0_l, Rminus_0_r, Rabs_pos_eq.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 = x
0 < eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 = x
0 <= 0
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 = x
forall i : nat,
(i <= n)%nat -> 0 = tg_alt (Ratan_seq 0) i
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
   assumption.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 = x
0 <= 0
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 = x
forall i : nat,
(i <= n)%nat -> 0 = tg_alt (Ratan_seq 0) i
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
  apply Rle_refl.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 = x
forall i : nat,
(i <= n)%nat -> 0 = tg_alt (Ratan_seq 0) i
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
 intros i _; unfold tg_alt, Ratan_seq, Rdiv; rewrite plus_comm; simpl.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 = x
i:nat
0 =
(-1) ^ i *
(0 * 0 ^ (i + (i + 0)) *
 /
 match (i + (i + 0))%nat with
 | 0%nat => 1
 | S _ => INR (i + (i + 0)) + 1
 end)
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
 solve[rewrite !Rmult_0_l, Rmult_0_r; auto].eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
replace (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) with
  (-(ps_atan (-x) - sum_f_R0 (tg_alt (Ratan_seq (-x))) n)).eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Rabs
  (-
   (ps_atan (- x) -
    sum_f_R0 (tg_alt (Ratan_seq (- x))) n)) < eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
-
(ps_atan (- x) - sum_f_R0 (tg_alt (Ratan_seq (- x))) n) =
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n
 rewrite Rabs_Ropp.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Rabs
  (ps_atan (- x) -
   sum_f_R0 (tg_alt (Ratan_seq (- x))) n) < eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
-
(ps_atan (- x) - sum_f_R0 (tg_alt (Ratan_seq (- x))) n) =
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n
 assert (Boule (/2) {| pos := / 2; cond_pos := pos_half_prf|} (-x)).eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
Boule (/ 2) {| pos := / 2; cond_pos := pos_half_prf |}
  (- x)
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
H:Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} 
  (- x)
Rabs
  (ps_atan (- x) -
   sum_f_R0 (tg_alt (Ratan_seq (- x))) n) < eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
-
(ps_atan (- x) - sum_f_R0 (tg_alt (Ratan_seq (- x))) n) =
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n
  revert b_y; unfold Boule; simpl; intros b_y; apply Rabs_def2 in b_y.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
x0:0 > x
b_y:x - 0 < 1 /\ - (1) < x - 0
Rabs (- x - / 2) < / 2
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
H:Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} 
  (- x)
Rabs
  (ps_atan (- x) -
   sum_f_R0 (tg_alt (Ratan_seq (- x))) n) < eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
-
(ps_atan (- x) - sum_f_R0 (tg_alt (Ratan_seq (- x))) n) =
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n
  destruct b_y; unfold Boule; simpl; apply Rabs_def1; lra.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
H:Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} 
  (- x)
Rabs
  (ps_atan (- x) -
   sum_f_R0 (tg_alt (Ratan_seq (- x))) n) < eps
eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
-
(ps_atan (- x) - sum_f_R0 (tg_alt (Ratan_seq (- x))) n) =
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n
 apply Pn; assumption.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
-
(ps_atan (- x) - sum_f_R0 (tg_alt (Ratan_seq (- x))) n) =
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n
unfold Rminus; rewrite ps_atan_opp, Ropp_plus_distr, sum_Ratan_seq_opp.eps:R
ep:0 < eps
N:nat
Pn:forall (n0 : nat) (y : R),
(N <= n0)%nat ->
Boule (/ 2)
  {| pos := / 2; cond_pos := pos_half_prf |} y ->
Rabs
  (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) <
eps
n:nat
x:R
nN:(N <= n)%nat
b_y:Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
x0:0 > x
- - ps_atan x +
- - - sum_f_R0 (tg_alt (Ratan_seq x)) n =
ps_atan x + - sum_f_R0 (tg_alt (Ratan_seq x)) n
rewrite !Ropp_involutive; reflexivity.
Qed.

Lemma Alt_PI_tg : forall n, PI_tg n = Ratan_seq 1 n.forall n : nat, PI_tg n = Ratan_seq 1 n
Proof.forall n : nat, PI_tg n = Ratan_seq 1 n
intros n; unfold PI_tg, Ratan_seq, Rdiv; rewrite pow1, Rmult_1_l.n:nat
/ INR (2 * n + 1) = / INR (2 * n + 1)
reflexivity.
Qed.

Lemma Ratan_is_ps_atan : forall eps, eps > 0 ->
       exists N, forall n, (n >= N)%nat -> forall x, -1 < x -> x < 1 ->
       Rabs (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) < eps.forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  forall x : R,
  -1 < x ->
  x < 1 ->
  Rabs (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
  eps
Proof.forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  forall x : R,
  -1 < x ->
  x < 1 ->
  Rabs (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
  eps
intros eps ep.eps:R
ep:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  forall x : R,
  -1 < x ->
  x < 1 ->
  Rabs (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
  eps
destruct (Ratan_CVU _ ep) as [N1 PN1].eps:R
ep:eps > 0
N1:nat
PN1:forall (n : nat) (y : R),
(N1 <= n)%nat ->
Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} y ->
Rabs (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n) < eps
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  forall x : R,
  -1 < x ->
  x < 1 ->
  Rabs (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
  eps
exists N1; intros n nN x xm1 x1; rewrite <- Rabs_Ropp, Ropp_minus_distr.eps:R
ep:eps > 0
N1:nat
PN1:forall (n0 : nat) (y : R),
(N1 <= n0)%nat ->
Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} y ->
Rabs (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) < eps
n:nat
nN:(n >= N1)%nat
x:R
xm1:-1 < x
x1:x < 1
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) <
eps
apply PN1; [assumption | ].eps:R
ep:eps > 0
N1:nat
PN1:forall (n0 : nat) (y : R),
(N1 <= n0)%nat ->
Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} y ->
Rabs (ps_atan y - sum_f_R0 (tg_alt (Ratan_seq y)) n0) < eps
n:nat
nN:(n >= N1)%nat
x:R
xm1:-1 < x
x1:x < 1
Boule 0 {| pos := 1; cond_pos := Rlt_0_1 |} x
unfold Boule; simpl; rewrite Rminus_0_r; apply Rabs_def1; assumption.
Qed.

Lemma Datan_continuity : continuity (fun x => /(1+x ^ 2)).continuity (fun x : R => / (1 + x ^ 2))
Proof.continuity (fun x : R => / (1 + x ^ 2))
apply continuity_inv.continuity (fun x : R => 1 + x ^ 2)
forall x : R, 1 + x ^ 2 <> 0
apply continuity_plus.continuity (fun _ : R => 1)
continuity (fun x : R => x ^ 2)
forall x : R, 1 + x ^ 2 <> 0
apply continuity_const ; unfold constant ; intuition.continuity (fun x : R => x ^ 2)
forall x : R, 1 + x ^ 2 <> 0
apply derivable_continuous ; apply derivable_pow.forall x : R, 1 + x ^ 2 <> 0
intro x ; apply Rgt_not_eq ; apply Rge_gt_trans with (1+0) ; [|lra] ;
 apply Rplus_ge_compat_l.x:R
x ^ 2 >= 0
 replace (x^2) with (x²).x:R
x² >= 0
x:R
x² = x ^ 2
 apply Rle_ge ; apply Rle_0_sqr.x:R
x² = x ^ 2
 unfold Rsqr ; field.
Qed.

Lemma derivable_pt_lim_ps_atan : forall x, -1 < x < 1 ->
  derivable_pt_lim ps_atan x ((fun y => /(1 + y ^ 2)) x).forall x : R,
-1 < x < 1 ->
derivable_pt_lim ps_atan x
  ((fun y : R => / (1 + y ^ 2)) x)
Proof.forall x : R,
-1 < x < 1 ->
derivable_pt_lim ps_atan x
  ((fun y : R => / (1 + y ^ 2)) x)
intros x x_encad.x:R
x_encad:-1 < x < 1
derivable_pt_lim ps_atan x
  ((fun y : R => / (1 + y ^ 2)) x)
destruct (boule_in_interval (-1) 1 x x_encad) as [c [r [Pcr1 [P1 P2]]]].x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
derivable_pt_lim ps_atan x (/ (1 + x ^ 2))
change (/ (1 + x ^ 2)) with ((fun u => /(1 + u ^ 2)) x).x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
derivable_pt_lim ps_atan x
  ((fun u : R => / (1 + u ^ 2)) x)
assert (t := derivable_pt_lim_CVU).x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
derivable_pt_lim ps_atan x
  ((fun u : R => / (1 + u ^ 2)) x)
apply derivable_pt_lim_CVU with 
          (fn := (fun N x => sum_f_R0 (tg_alt (Ratan_seq x)) N))
          (fn' := (fun N x => sum_f_R0 (tg_alt (Datan_seq x)) N))
          (c := c) (r := r).x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
Boule c r x
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall (y : R) (n : nat),
Boule c r y ->
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) n) y
  (sum_f_R0 (tg_alt (Datan_seq y)) n)
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
Un_cv
  (fun n : nat => sum_f_R0 (tg_alt (Ratan_seq y)) n)
  (ps_atan y)
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
CVU
  (fun (N : nat) (x0 : R) =>
   sum_f_R0 (tg_alt (Datan_seq x0)) N)
  (fun u : R => / (1 + u ^ 2)) c r
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
    assumption.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall (y : R) (n : nat),
Boule c r y ->
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) n) y
  (sum_f_R0 (tg_alt (Datan_seq y)) n)
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
Un_cv
  (fun n : nat => sum_f_R0 (tg_alt (Ratan_seq y)) n)
  (ps_atan y)
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
CVU
  (fun (N : nat) (x0 : R) =>
   sum_f_R0 (tg_alt (Datan_seq x0)) N)
  (fun u : R => / (1 + u ^ 2)) c r
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
   intros y N inb; apply Rabs_def2 in inb; destruct inb.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y0 : R) (n : nat),
 Boule c0 r0 y0 ->
 derivable_pt_lim (fn n) y0 (fn' n y0)) ->
(forall y0 : R,
 Boule c0 r0 y0 ->
 Un_cv (fun n : nat => fn n y0) (f y0)) ->
CVU fn' g c0 r0 ->
(forall y0 : R,
 Boule c0 r0 y0 -> continuity_pt g y0) ->
derivable_pt_lim f x0 (g x0)
y:R
N:nat
H:y - c < r
H0:- r < y - c
derivable_pt_lim
  (fun x0 : R => sum_f_R0 (tg_alt (Ratan_seq x0)) N) y
  (sum_f_R0 (tg_alt (Datan_seq y)) N)
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
Un_cv
  (fun n : nat => sum_f_R0 (tg_alt (Ratan_seq y)) n)
  (ps_atan y)
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
CVU
  (fun (N : nat) (x0 : R) =>
   sum_f_R0 (tg_alt (Datan_seq x0)) N)
  (fun u : R => / (1 + u ^ 2)) c r
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
   apply Datan_is_datan.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y0 : R) (n : nat),
 Boule c0 r0 y0 ->
 derivable_pt_lim (fn n) y0 (fn' n y0)) ->
(forall y0 : R,
 Boule c0 r0 y0 ->
 Un_cv (fun n : nat => fn n y0) (f y0)) ->
CVU fn' g c0 r0 ->
(forall y0 : R,
 Boule c0 r0 y0 -> continuity_pt g y0) ->
derivable_pt_lim f x0 (g x0)
y:R
N:nat
H:y - c < r
H0:- r < y - c
-1 <= y
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y0 : R) (n : nat),
 Boule c0 r0 y0 ->
 derivable_pt_lim (fn n) y0 (fn' n y0)) ->
(forall y0 : R,
 Boule c0 r0 y0 ->
 Un_cv (fun n : nat => fn n y0) (f y0)) ->
CVU fn' g c0 r0 ->
(forall y0 : R,
 Boule c0 r0 y0 -> continuity_pt g y0) ->
derivable_pt_lim f x0 (g x0)
y:R
N:nat
H:y - c < r
H0:- r < y - c
y < 1
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
Un_cv
  (fun n : nat => sum_f_R0 (tg_alt (Ratan_seq y)) n)
  (ps_atan y)
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
CVU
  (fun (N : nat) (x0 : R) =>
   sum_f_R0 (tg_alt (Datan_seq x0)) N)
  (fun u : R => / (1 + u ^ 2)) c r
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
    lra.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y0 : R) (n : nat),
 Boule c0 r0 y0 ->
 derivable_pt_lim (fn n) y0 (fn' n y0)) ->
(forall y0 : R,
 Boule c0 r0 y0 ->
 Un_cv (fun n : nat => fn n y0) (f y0)) ->
CVU fn' g c0 r0 ->
(forall y0 : R,
 Boule c0 r0 y0 -> continuity_pt g y0) ->
derivable_pt_lim f x0 (g x0)
y:R
N:nat
H:y - c < r
H0:- r < y - c
y < 1
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
Un_cv
  (fun n : nat => sum_f_R0 (tg_alt (Ratan_seq y)) n)
  (ps_atan y)
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
CVU
  (fun (N : nat) (x0 : R) =>
   sum_f_R0 (tg_alt (Datan_seq x0)) N)
  (fun u : R => / (1 + u ^ 2)) c r
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
   lra.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
Un_cv
  (fun n : nat => sum_f_R0 (tg_alt (Ratan_seq y)) n)
  (ps_atan y)
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
CVU
  (fun (N : nat) (x0 : R) =>
   sum_f_R0 (tg_alt (Datan_seq x0)) N)
  (fun u : R => / (1 + u ^ 2)) c r
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
  intros y inb; apply Rabs_def2 in inb; destruct inb.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y0 : R) (n : nat),
 Boule c0 r0 y0 ->
 derivable_pt_lim (fn n) y0 (fn' n y0)) ->
(forall y0 : R,
 Boule c0 r0 y0 ->
 Un_cv (fun n : nat => fn n y0) (f y0)) ->
CVU fn' g c0 r0 ->
(forall y0 : R,
 Boule c0 r0 y0 -> continuity_pt g y0) ->
derivable_pt_lim f x0 (g x0)
y:R
H:y - c < r
H0:- r < y - c
Un_cv
  (fun n : nat => sum_f_R0 (tg_alt (Ratan_seq y)) n)
  (ps_atan y)
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
CVU
  (fun (N : nat) (x0 : R) =>
   sum_f_R0 (tg_alt (Datan_seq x0)) N)
  (fun u : R => / (1 + u ^ 2)) c r
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
  assert (y_gt_0 : -1 < y) by lra.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y0 : R) (n : nat),
 Boule c0 r0 y0 ->
 derivable_pt_lim (fn n) y0 (fn' n y0)) ->
(forall y0 : R,
 Boule c0 r0 y0 ->
 Un_cv (fun n : nat => fn n y0) (f y0)) ->
CVU fn' g c0 r0 ->
(forall y0 : R,
 Boule c0 r0 y0 -> continuity_pt g y0) ->
derivable_pt_lim f x0 (g x0)
y:R
H:y - c < r
H0:- r < y - c
y_gt_0:-1 < y
Un_cv
  (fun n : nat => sum_f_R0 (tg_alt (Ratan_seq y)) n)
  (ps_atan y)
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
CVU
  (fun (N : nat) (x0 : R) =>
   sum_f_R0 (tg_alt (Datan_seq x0)) N)
  (fun u : R => / (1 + u ^ 2)) c r
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
  assert (y_lt_1 : y < 1) by lra.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y0 : R) (n : nat),
 Boule c0 r0 y0 ->
 derivable_pt_lim (fn n) y0 (fn' n y0)) ->
(forall y0 : R,
 Boule c0 r0 y0 ->
 Un_cv (fun n : nat => fn n y0) (f y0)) ->
CVU fn' g c0 r0 ->
(forall y0 : R,
 Boule c0 r0 y0 -> continuity_pt g y0) ->
derivable_pt_lim f x0 (g x0)
y:R
H:y - c < r
H0:- r < y - c
y_gt_0:-1 < y
y_lt_1:y < 1
Un_cv
  (fun n : nat => sum_f_R0 (tg_alt (Ratan_seq y)) n)
  (ps_atan y)
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
CVU
  (fun (N : nat) (x0 : R) =>
   sum_f_R0 (tg_alt (Datan_seq x0)) N)
  (fun u : R => / (1 + u ^ 2)) c r
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
  intros eps eps_pos ; elim (Ratan_is_ps_atan eps eps_pos).x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y0 : R) (n : nat),
 Boule c0 r0 y0 ->
 derivable_pt_lim (fn n) y0 (fn' n y0)) ->
(forall y0 : R,
 Boule c0 r0 y0 ->
 Un_cv (fun n : nat => fn n y0) (f y0)) ->
CVU fn' g c0 r0 ->
(forall y0 : R,
 Boule c0 r0 y0 -> continuity_pt g y0) ->
derivable_pt_lim f x0 (g x0)
y:R
H:y - c < r
H0:- r < y - c
y_gt_0:-1 < y
y_lt_1:y < 1
eps:R
eps_pos:eps > 0
forall x0 : nat,
(forall n : nat,
 (n >= x0)%nat ->
 forall x1 : R,
 -1 < x1 ->
 x1 < 1 ->
 Rabs
   (sum_f_R0 (tg_alt (Ratan_seq x1)) n - ps_atan x1) <
 eps) ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (tg_alt (Ratan_seq y)) n)
    (ps_atan y) < eps
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
CVU
  (fun (N : nat) (x0 : R) =>
   sum_f_R0 (tg_alt (Datan_seq x0)) N)
  (fun u : R => / (1 + u ^ 2)) c r
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
  intros N HN ; exists N; intros n n_lb ; apply HN ; tauto.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
CVU
  (fun (N : nat) (x0 : R) =>
   sum_f_R0 (tg_alt (Datan_seq x0)) N)
  (fun u : R => / (1 + u ^ 2)) c r
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
 apply Datan_CVU_prelim.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
Rabs c + r < 1
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
 replace ((c - r + (c + r)) / 2) with c by field.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
Rabs c + r < 1
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
 unfold mkposreal_lb_ub; simpl.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
Rabs c + r < 1
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
 replace ((c + r - (c - r)) / 2) with (r :R) by field.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
Rabs c + r < 1
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
 assert (Rabs c < 1 - r).x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
Rabs c < 1 - r
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
H:Rabs c < 1 - r
Rabs c + r < 1
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
  unfold Boule in Pcr1; destruct r; simpl in *; apply Rabs_def1;
  apply Rabs_def2 in Pcr1; destruct Pcr1; lra.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
H:Rabs c < 1 - r
Rabs c + r < 1
x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
 lra.x:R
x_encad:-1 < x < 1
c:R
r:posreal
Pcr1:Boule c r x
P1:-1 < c - r
P2:c + r < 1
t:forall (fn fn' : nat -> R -> R) 
  (f g : R -> R) (x0 c0 : R) 
  (r0 : posreal),
Boule c0 r0 x0 ->
(forall (y : R) (n : nat),
 Boule c0 r0 y ->
 derivable_pt_lim (fn n) y (fn' n y)) ->
(forall y : R,
 Boule c0 r0 y ->
 Un_cv (fun n : nat => fn n y) (f y)) ->
CVU fn' g c0 r0 ->
(forall y : R, Boule c0 r0 y -> continuity_pt g y) ->
derivable_pt_lim f x0 (g x0)
forall y : R,
Boule c r y ->
continuity_pt (fun u : R => / (1 + u ^ 2)) y
intros; apply Datan_continuity.
Qed.

Lemma derivable_pt_ps_atan :
   forall x, -1 < x < 1 -> derivable_pt ps_atan x.forall x : R, -1 < x < 1 -> derivable_pt ps_atan x
Proof.forall x : R, -1 < x < 1 -> derivable_pt ps_atan x
intros x x_encad.x:R
x_encad:-1 < x < 1
derivable_pt ps_atan x
exists (/(1+x^2)) ; apply derivable_pt_lim_ps_atan; assumption.
Qed.

Lemma ps_atan_continuity_pt_1 : forall eps : R,
       eps > 0 ->
       exists alp : R,
       alp > 0 /\
       (forall x, x < 1 -> 0 < x -> R_dist x 1 < alp ->
       dist R_met (ps_atan x) (Alt_PI/4) < eps).forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   x < 1 ->
   0 < x ->
   R_dist x 1 < alp ->
   dist R_met (ps_atan x) (Alt_PI / 4) < eps)
Proof.forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   x < 1 ->
   0 < x ->
   R_dist x 1 < alp ->
   dist R_met (ps_atan x) (Alt_PI / 4) < eps)
intros eps eps_pos.eps:R
eps_pos:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x : R,
   x < 1 ->
   0 < x ->
   R_dist x 1 < alp ->
   dist R_met (ps_atan x) (Alt_PI / 4) < eps)
assert (eps_3_pos : eps / 3 > 0) by lra.eps:R
eps_pos:eps > 0
eps_3_pos:eps / 3 > 0
exists alp : R,
  alp > 0 /\
  (forall x : R,
   x < 1 ->
   0 < x ->
   R_dist x 1 < alp ->
   dist R_met (ps_atan x) (Alt_PI / 4) < eps)
elim (Ratan_is_ps_atan (eps / 3) eps_3_pos) ; intros N1 HN1.eps:R
eps_pos:eps > 0
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x : R,
-1 < x ->
x < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
eps / 3
exists alp : R,
  alp > 0 /\
  (forall x : R,
   x < 1 ->
   0 < x ->
   R_dist x 1 < alp ->
   dist R_met (ps_atan x) (Alt_PI / 4) < eps)
unfold Alt_PI.eps:R
eps_pos:eps > 0
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x : R,
-1 < x ->
x < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
eps / 3
exists alp : R,
  alp > 0 /\
  (forall x : R,
   x < 1 ->
   0 < x ->
   R_dist x 1 < alp ->
   dist R_met (ps_atan x)
     (4 * (let (a, _) := exist_PI in a) / 4) < eps)
destruct exist_PI as [v Pv]; replace ((4 * v)/4) with v by field.eps:R
eps_pos:eps > 0
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x : R,
-1 < x ->
x < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
eps / 3
v:R
Pv:Un_cv (fun N : nat => sum_f_R0 (tg_alt PI_tg) N)
  v
exists alp : R,
  alp > 0 /\
  (forall x : R,
   x < 1 ->
   0 < x ->
   R_dist x 1 < alp -> dist R_met (ps_atan x) v < eps)
assert (Pv' : Un_cv (sum_f_R0 (tg_alt (Ratan_seq 1))) v).eps:R
eps_pos:eps > 0
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x : R,
-1 < x ->
x < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
eps / 3
v:R
Pv:Un_cv (fun N : nat => sum_f_R0 (tg_alt PI_tg) N)
  v
Un_cv (sum_f_R0 (tg_alt (Ratan_seq 1))) v
eps:R
eps_pos:eps > 0
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x : R,
-1 < x ->
x < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
eps / 3
v:R
Pv:Un_cv (fun N : nat => sum_f_R0 (tg_alt PI_tg) N)
  v
Pv':Un_cv (sum_f_R0 (tg_alt (Ratan_seq 1))) v
exists alp : R,
  alp > 0 /\
  (forall x : R,
   x < 1 ->
   0 < x ->
   R_dist x 1 < alp -> dist R_met (ps_atan x) v < eps)
 apply Un_cv_ext with (2:= Pv).eps:R
eps_pos:eps > 0
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x : R,
-1 < x ->
x < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
eps / 3
v:R
Pv:Un_cv (fun N : nat => sum_f_R0 (tg_alt PI_tg) N)
  v
forall n : nat,
sum_f_R0 (tg_alt PI_tg) n =
sum_f_R0 (tg_alt (Ratan_seq 1)) n
eps:R
eps_pos:eps > 0
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x : R,
-1 < x ->
x < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
eps / 3
v:R
Pv:Un_cv (fun N : nat => sum_f_R0 (tg_alt PI_tg) N)
  v
Pv':Un_cv (sum_f_R0 (tg_alt (Ratan_seq 1))) v
exists alp : R,
  alp > 0 /\
  (forall x : R,
   x < 1 ->
   0 < x ->
   R_dist x 1 < alp -> dist R_met (ps_atan x) v < eps)
 intros; apply sum_eq; intros; unfold tg_alt; rewrite Alt_PI_tg; tauto.eps:R
eps_pos:eps > 0
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x : R,
-1 < x ->
x < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
eps / 3
v:R
Pv:Un_cv (fun N : nat => sum_f_R0 (tg_alt PI_tg) N)
  v
Pv':Un_cv (sum_f_R0 (tg_alt (Ratan_seq 1))) v
exists alp : R,
  alp > 0 /\
  (forall x : R,
   x < 1 ->
   0 < x ->
   R_dist x 1 < alp -> dist R_met (ps_atan x) v < eps)
destruct (Pv' (eps / 3) eps_3_pos) as [N2 HN2].eps:R
eps_pos:eps > 0
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x : R,
-1 < x ->
x < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
eps / 3
v:R
Pv:Un_cv (fun N : nat => sum_f_R0 (tg_alt PI_tg) N)
  v
Pv':Un_cv (sum_f_R0 (tg_alt (Ratan_seq 1))) v
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
exists alp : R,
  alp > 0 /\
  (forall x : R,
   x < 1 ->
   0 < x ->
   R_dist x 1 < alp -> dist R_met (ps_atan x) v < eps)
set (N := (N1 + N2)%nat).eps:R
eps_pos:eps > 0
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x : R,
-1 < x ->
x < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
eps / 3
v:R
Pv:Un_cv
  (fun N0 : nat => sum_f_R0 (tg_alt PI_tg) N0) v
Pv':Un_cv (sum_f_R0 (tg_alt (Ratan_seq 1))) v
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
exists alp : R,
  alp > 0 /\
  (forall x : R,
   x < 1 ->
   0 < x ->
   R_dist x 1 < alp -> dist R_met (ps_atan x) v < eps)
assert (O_lb : 0 <= 1) by intuition ; assert (O_ub : 1 <= 1) by intuition ;
 elim (ps_atanSeq_continuity_pt_1 N 1 O_lb O_ub (eps / 3) eps_3_pos) ; intros alpha Halpha ;
 clear -HN1 HN2 Halpha eps_3_pos; destruct Halpha as (alpha_pos, Halpha).eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x : R,
-1 < x ->
x < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
exists alp : R,
  alp > 0 /\
  (forall x : R,
   x < 1 ->
   0 < x ->
   R_dist x 1 < alp -> dist R_met (ps_atan x) v < eps)
exists alpha ; split;[assumption | ].eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x : R,
-1 < x ->
x < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) <
eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha -> dist R_met (ps_atan x) v < eps
intros x x_ub x_lb x_bounds.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
dist R_met (ps_atan x) v < eps
simpl ; unfold R_dist.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - v) < eps
replace (ps_atan x - v) with ((ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N)
     + (sum_f_R0 (tg_alt (Ratan_seq x)) N - sum_f_R0 (tg_alt (Ratan_seq 1)) N)
     + (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v)).eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs
  (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
   (sum_f_R0 (tg_alt (Ratan_seq x)) N -
    sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
   (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v)) < eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
apply Rle_lt_trans with (r2:=Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) +
      Rabs ((sum_f_R0 (tg_alt (Ratan_seq x)) N - sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
   (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v))).eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs
  (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
   (sum_f_R0 (tg_alt (Ratan_seq x)) N -
    sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
   (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v)) <=
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) +
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N +
   (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v))
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) +
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N +
   (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v)) < eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
rewrite Rplus_assoc ; apply Rabs_triang.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) +
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N +
   (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v)) < eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
   replace eps with (2 / 3 * eps + eps / 3).eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) +
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N +
   (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v)) <
2 / 3 * eps + eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
   rewrite Rplus_comm.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N +
   (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v)) +
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) <
2 / 3 * eps + eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
   apply Rplus_lt_compat.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N +
   (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v)) <
2 / 3 * eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) <
eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
   apply Rle_lt_trans with (r2 := Rabs (sum_f_R0 (tg_alt (Ratan_seq x)) N - sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
      Rabs (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v)).eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N +
   (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v)) <=
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
Rabs (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v)
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
Rabs (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) <
2 / 3 * eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) <
eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
   apply Rabs_triang.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
Rabs (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) <
2 / 3 * eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) <
eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
   apply Rlt_le_trans with (r2:= eps / 3 + eps / 3).eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
Rabs (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) <
eps / 3 + eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
eps / 3 + eps / 3 <= 2 / 3 * eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) <
eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
   apply Rplus_lt_compat.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N) < 
eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) < eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
eps / 3 + eps / 3 <= 2 / 3 * eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) <
eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
   simpl in Halpha ; unfold R_dist in Halpha.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alpha ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N) < 
eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) < eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
eps / 3 + eps / 3 <= 2 / 3 * eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) <
eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
   apply Halpha ; split.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alpha ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
D_x no_cond 1 x
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alpha ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (x - 1) < alpha
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) < eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
eps / 3 + eps / 3 <= 2 / 3 * eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) <
eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
   unfold D_x, no_cond ; split ; [ | apply Rgt_not_eq ] ; intuition.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alpha ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N -
   sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (x - 1) < alpha
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) < eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
eps / 3 + eps / 3 <= 2 / 3 * eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) <
eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
   intuition.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) < eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
eps / 3 + eps / 3 <= 2 / 3 * eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) <
eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
   apply HN2; unfold N; omega.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
eps / 3 + eps / 3 <= 2 / 3 * eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) <
eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
   lra.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) <
eps / 3
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
  rewrite <- Rabs_Ropp, Ropp_minus_distr; apply HN1.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
(N >= N1)%nat
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
-1 < x
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
x < 1
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
     unfold N; omega.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
-1 < x
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
x < 1
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v  
    lra.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
x < 1
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
  assumption.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
2 / 3 * eps + eps / 3 = eps
eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
 field.eps:R
eps_3_pos:eps / 3 > 0
N1:nat
HN1:forall n : nat,
(n >= N1)%nat ->
forall x0 : R,
-1 < x0 ->
x0 < 1 ->
Rabs
  (sum_f_R0 (tg_alt (Ratan_seq x0)) n -
   ps_atan x0) < eps / 3
v:R
N2:nat
HN2:forall n : nat,
(n >= N2)%nat ->
R_dist (sum_f_R0 (tg_alt (Ratan_seq 1)) n) v <
eps / 3
N:=(N1 + N2)%nat:nat
alpha:R
alpha_pos:alpha > 0
Halpha:forall x0 : Base R_met,
D_x no_cond 1 x0 /\ dist R_met x0 1 < alpha ->
dist R_met
  (sum_f_R0 (tg_alt (Ratan_seq x0)) N)
  (sum_f_R0 (tg_alt (Ratan_seq 1)) N) <
eps / 3
x:R
x_ub:x < 1
x_lb:0 < x
x_bounds:R_dist x 1 < alpha
ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N +
(sum_f_R0 (tg_alt (Ratan_seq x)) N -
 sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
(sum_f_R0 (tg_alt (Ratan_seq 1)) N - v) =
ps_atan x - v
ring.
Qed.

Lemma Datan_eq_DatanSeq_interv : forall x, -1 < x < 1 ->
 forall (Pratan:derivable_pt ps_atan x) (Prmymeta:derivable_pt atan x),
      derive_pt ps_atan x Pratan = derive_pt atan x Prmymeta.forall x : R,
-1 < x < 1 ->
forall (Pratan : derivable_pt ps_atan x)
  (Prmymeta : derivable_pt atan x),
derive_pt ps_atan x Pratan = derive_pt atan x Prmymeta
Proof.forall x : R,
-1 < x < 1 ->
forall (Pratan : derivable_pt ps_atan x)
  (Prmymeta : derivable_pt atan x),
derive_pt ps_atan x Pratan = derive_pt atan x Prmymeta
assert (freq : 0 < tan 1) by apply (Rlt_trans _ _ _ Rlt_0_1 tan_1_gt_1).freq:0 < tan 1
forall x : R,
-1 < x < 1 ->
forall (Pratan : derivable_pt ps_atan x)
  (Prmymeta : derivable_pt atan x),
derive_pt ps_atan x Pratan = derive_pt atan x Prmymeta
intros x x_encad Pratan Prmymeta.freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
derive_pt ps_atan x Pratan = derive_pt atan x Prmymeta
 rewrite pr_nu_var2_interv with (g:=ps_atan) (lb:=-1) (ub:=tan 1)
   (pr2 := derivable_pt_ps_atan x x_encad).freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
derive_pt ps_atan x (derivable_pt_ps_atan x x_encad) =
derive_pt atan x Prmymeta
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
 rewrite pr_nu_var2_interv with (f:=atan) (g:=atan) (lb:=-1) (ub:= 1) (pr2:=derivable_pt_atan x).freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
derive_pt ps_atan x (derivable_pt_ps_atan x x_encad) =
derive_pt atan x (derivable_pt_atan x)
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < 1 -> atan h = atan h
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
 assert (Temp := derivable_pt_lim_ps_atan x x_encad).freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
Temp:derivable_pt_lim ps_atan x
  ((fun y : R => / (1 + y ^ 2)) x)
derive_pt ps_atan x (derivable_pt_ps_atan x x_encad) =
derive_pt atan x (derivable_pt_atan x)
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < 1 -> atan h = atan h
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
 assert (Hrew1 : derive_pt ps_atan x (derivable_pt_ps_atan x x_encad) = (/(1+x^2))).freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
Temp:derivable_pt_lim ps_atan x
  ((fun y : R => / (1 + y ^ 2)) x)
derive_pt ps_atan x (derivable_pt_ps_atan x x_encad) =
/ (1 + x ^ 2)
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
Temp:derivable_pt_lim ps_atan x
  ((fun y : R => / (1 + y ^ 2)) x)
Hrew1:derive_pt ps_atan x
  (derivable_pt_ps_atan x x_encad) =
/ (1 + x ^ 2)
derive_pt ps_atan x (derivable_pt_ps_atan x x_encad) =
derive_pt atan x (derivable_pt_atan x)
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < 1 -> atan h = atan h
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
  apply derive_pt_eq_0 ; assumption.freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
Temp:derivable_pt_lim ps_atan x
  ((fun y : R => / (1 + y ^ 2)) x)
Hrew1:derive_pt ps_atan x
  (derivable_pt_ps_atan x x_encad) =
/ (1 + x ^ 2)
derive_pt ps_atan x (derivable_pt_ps_atan x x_encad) =
derive_pt atan x (derivable_pt_atan x)
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < 1 -> atan h = atan h
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
  rewrite derive_pt_atan.freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
Temp:derivable_pt_lim ps_atan x
  ((fun y : R => / (1 + y ^ 2)) x)
Hrew1:derive_pt ps_atan x
  (derivable_pt_ps_atan x x_encad) =
/ (1 + x ^ 2)
derive_pt ps_atan x (derivable_pt_ps_atan x x_encad) =
1 / (1 + x²)
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < 1 -> atan h = atan h
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
  rewrite Hrew1.freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
Temp:derivable_pt_lim ps_atan x
  ((fun y : R => / (1 + y ^ 2)) x)
Hrew1:derive_pt ps_atan x
  (derivable_pt_ps_atan x x_encad) =
/ (1 + x ^ 2)
/ (1 + x ^ 2) = 1 / (1 + x²)
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < 1 -> atan h = atan h
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
  replace (Rsqr x) with (x ^ 2) by (unfold Rsqr; ring).freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
Temp:derivable_pt_lim ps_atan x
  ((fun y : R => / (1 + y ^ 2)) x)
Hrew1:derive_pt ps_atan x
  (derivable_pt_ps_atan x x_encad) =
/ (1 + x ^ 2)
/ (1 + x ^ 2) = 1 / (1 + x ^ 2)
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < 1 -> atan h = atan h
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
  unfold Rdiv; rewrite Rmult_1_l; reflexivity.freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < 1 -> atan h = atan h
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
     lra.freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < 1 -> atan h = atan h
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
    assumption.freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < 1 -> atan h = atan h
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
   intros; reflexivity.freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
  lra.freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
-1 < x < tan 1
freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
 assert (t := tan_1_gt_1); split;destruct x_encad; lra.freq:0 < tan 1
x:R
x_encad:-1 < x < 1
Pratan:derivable_pt ps_atan x
Prmymeta:derivable_pt atan x
forall h : R, -1 < h < tan 1 -> ps_atan h = ps_atan h
intros; reflexivity.
Qed.

Lemma atan_eq_ps_atan :
 forall x, 0 < x < 1 -> atan x = ps_atan x.forall x : R, 0 < x < 1 -> atan x = ps_atan x
Proof.forall x : R, 0 < x < 1 -> atan x = ps_atan x
intros x x_encad.x:R
x_encad:0 < x < 1
atan x = ps_atan x
assert (pr1 : forall c : R, 0 < c < x -> derivable_pt (atan - ps_atan) c).x:R
x_encad:0 < x < 1
forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
atan x = ps_atan x
 intros c c_encad.x:R
x_encad:0 < x < 1
c:R
c_encad:0 < c < x
derivable_pt (atan - ps_atan) c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
atan x = ps_atan x
 apply derivable_pt_minus.x:R
x_encad:0 < x < 1
c:R
c_encad:0 < c < x
derivable_pt atan c
x:R
x_encad:0 < x < 1
c:R
c_encad:0 < c < x
derivable_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
atan x = ps_atan x
  exact (derivable_pt_atan c).x:R
x_encad:0 < x < 1
c:R
c_encad:0 < c < x
derivable_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
atan x = ps_atan x
 apply derivable_pt_ps_atan.x:R
x_encad:0 < x < 1
c:R
c_encad:0 < c < x
-1 < c < 1
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
atan x = ps_atan x
  destruct x_encad; destruct c_encad; split; lra.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
atan x = ps_atan x
assert (pr2 : forall c : R, 0 < c < x -> derivable_pt id c).x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
forall c : R, 0 < c < x -> derivable_pt id c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
atan x = ps_atan x
 intros ; apply derivable_pt_id; lra.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
atan x = ps_atan x
assert (delta_cont : forall c : R, 0 <= c <= x -> continuity_pt (atan - ps_atan) c).x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
forall c : R,
0 <= c <= x -> continuity_pt (atan - ps_atan) c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x
 intros c [[c_encad1 | c_encad1 ] [c_encad2 | c_encad2]];
        apply continuity_pt_minus.x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c < x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c < x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x
        apply derivable_continuous_pt ; apply derivable_pt_atan.x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c < x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x 
       apply derivable_continuous_pt ; apply derivable_pt_ps_atan.x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c < x
-1 < c < 1
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x
       split; destruct x_encad; lra.x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x
      apply derivable_continuous_pt, derivable_pt_atan.x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x
     apply derivable_continuous_pt, derivable_pt_ps_atan.x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 < c
c_encad2:c = x
-1 < c < 1
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x
     subst c; destruct x_encad; split; lra.x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x
    apply derivable_continuous_pt, derivable_pt_atan.x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x
   apply derivable_continuous_pt, derivable_pt_ps_atan.x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c < x
-1 < c < 1
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x
   subst c; split; lra.x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt atan c
x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x
  apply derivable_continuous_pt, derivable_pt_atan.x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
continuity_pt ps_atan c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x
 apply derivable_continuous_pt, derivable_pt_ps_atan.x:R
x_encad:0 < x < 1
pr1:forall c0 : R,
0 < c0 < x -> derivable_pt (atan - ps_atan) c0
pr2:forall c0 : R, 0 < c0 < x -> derivable_pt id c0
c:R
c_encad1:0 = c
c_encad2:c = x
-1 < c < 1
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x
 subst c; destruct x_encad; split; lra.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
atan x = ps_atan x
assert (id_cont : forall c : R, 0 <= c <= x -> continuity_pt id c).x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
forall c : R, 0 <= c <= x -> continuity_pt id c
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
id_cont:forall c : R,
0 <= c <= x -> continuity_pt id c
atan x = ps_atan x
    intros ; apply derivable_continuous ; apply derivable_id.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
id_cont:forall c : R,
0 <= c <= x -> continuity_pt id c
atan x = ps_atan x
assert (x_lb : 0 < x) by (destruct x_encad; lra).x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
id_cont:forall c : R,
0 <= c <= x -> continuity_pt id c
x_lb:0 < x
atan x = ps_atan x
elim (MVT (atan - ps_atan)%F id 0 x pr1 pr2 x_lb delta_cont id_cont) ; intros d Temp ; elim Temp ; intros d_encad Main.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
delta_cont:forall c : R,
0 <= c <= x ->
continuity_pt (atan - ps_atan) c
id_cont:forall c : R,
0 <= c <= x -> continuity_pt id c
x_lb:0 < x
d:R
Temp:exists P : 0 < d < x,
  (id x - id 0) *
  derive_pt (atan - ps_atan) d (pr1 d P) =
  ((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
  derive_pt id d (pr2 d P)
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
atan x = ps_atan x
clear - Main x_encad.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
atan x = ps_atan x
assert (Temp : forall (pr: derivable_pt (atan - ps_atan) d), derive_pt (atan - ps_atan) d pr = 0).x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x
 intro pr.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
derive_pt (atan - ps_atan) d pr = 0
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x
 assert (d_encad3 : -1 < d < 1).x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
-1 < d < 1
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
derive_pt (atan - ps_atan) d pr = 0
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x
  destruct d_encad; destruct x_encad; split; lra.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
derive_pt (atan - ps_atan) d pr = 0
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x
 pose (pr3 := derivable_pt_minus atan ps_atan d (derivable_pt_atan d) (derivable_pt_ps_atan d d_encad3)).x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
derive_pt (atan - ps_atan) d pr = 0
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x
 rewrite <- pr_nu_var2_interv with (f:=(atan - ps_atan)%F) (g:=(atan - ps_atan)%F) (lb:=0) (ub:=x) (pr1:=pr3) (pr2:=pr).x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
derive_pt (atan - ps_atan) d pr3 = 0
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
0 < x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
0 < d < x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
forall h : R,
0 < h < x ->
(atan - ps_atan)%F h = (atan - ps_atan)%F h
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x
    unfold pr3.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
derive_pt (atan - ps_atan) d
  (derivable_pt_minus atan ps_atan d
     (derivable_pt_atan d)
     (derivable_pt_ps_atan d d_encad3)) = 0
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
0 < x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
0 < d < x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
forall h : R,
0 < h < x ->
(atan - ps_atan)%F h = (atan - ps_atan)%F h
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x rewrite derive_pt_minus.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
derive_pt atan d (derivable_pt_atan d) -
derive_pt ps_atan d (derivable_pt_ps_atan d d_encad3) =
0
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
0 < x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
0 < d < x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
forall h : R,
0 < h < x ->
(atan - ps_atan)%F h = (atan - ps_atan)%F h
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x
    rewrite Datan_eq_DatanSeq_interv with (Prmymeta := derivable_pt_atan d).x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
derive_pt atan d (derivable_pt_atan d) -
derive_pt atan d (derivable_pt_atan d) = 0
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
-1 < d < 1
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
0 < x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
0 < d < x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
forall h : R,
0 < h < x ->
(atan - ps_atan)%F h = (atan - ps_atan)%F h
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x
     intuition.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
-1 < d < 1
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
0 < x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
0 < d < x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
forall h : R,
0 < h < x ->
(atan - ps_atan)%F h = (atan - ps_atan)%F h
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x
    assumption.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
0 < x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
0 < d < x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
forall h : R,
0 < h < x ->
(atan - ps_atan)%F h = (atan - ps_atan)%F h
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x 
   destruct d_encad; lra.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
0 < d < x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
forall h : R,
0 < h < x ->
(atan - ps_atan)%F h = (atan - ps_atan)%F h
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x
  assumption.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
pr:derivable_pt (atan - ps_atan) d
d_encad3:-1 < d < 1
pr3:=derivable_pt_minus atan ps_atan d
  (derivable_pt_atan d)
  (derivable_pt_ps_atan d d_encad3):derivable_pt (atan - ps_atan) d
forall h : R,
0 < h < x ->
(atan - ps_atan)%F h = (atan - ps_atan)%F h
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x
 reflexivity.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan x = ps_atan x
assert (iatan0 : atan 0 = 0).x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
atan 0 = 0
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
atan x = ps_atan x
 apply tan_is_inj.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
- PI / 2 < atan 0 < PI / 2
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
- PI / 2 < 0 < PI / 2
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
tan (atan 0) = tan 0
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
atan x = ps_atan x
   apply atan_bound.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
- PI / 2 < 0 < PI / 2
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
tan (atan 0) = tan 0
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
atan x = ps_atan x
  rewrite Ropp_div; assert (t := PI2_RGT_0); split; lra.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
tan (atan 0) = tan 0
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
atan x = ps_atan x
 rewrite tan_0, atan_right_inv; reflexivity.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
atan x = ps_atan x
generalize Main; rewrite Temp, Rmult_0_r.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
0 =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad) -> 
atan x = ps_atan x
replace ((atan - ps_atan)%F x) with (atan x - ps_atan x) by intuition.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
0 =
(atan x - ps_atan x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad) -> 
atan x = ps_atan x
replace ((atan - ps_atan)%F 0) with (atan 0 - ps_atan 0) by intuition.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
0 =
(atan x - ps_atan x - (atan 0 - ps_atan 0)) *
derive_pt id d (pr2 d d_encad) -> 
atan x = ps_atan x
rewrite iatan0, ps_atan0_0, !Rminus_0_r.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
0 =
(atan x - ps_atan x) * derive_pt id d (pr2 d d_encad) ->
atan x = ps_atan x
replace (derive_pt id d (pr2 d d_encad)) with 1.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
0 = (atan x - ps_atan x) * 1 -> atan x = ps_atan x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
1 = derive_pt id d (pr2 d d_encad) 
 rewrite Rmult_1_r.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
0 = atan x - ps_atan x -> atan x = ps_atan x
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
1 = derive_pt id d (pr2 d d_encad)
 solve[intros M; apply Rminus_diag_uniq; auto].x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
1 = derive_pt id d (pr2 d d_encad)
rewrite pr_nu_var with (g:=id) (pr2:=derivable_pt_id d).x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
1 = derive_pt id d (derivable_pt_id d)
x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
id = id
 symmetry ; apply derive_pt_id.x:R
x_encad:0 < x < 1
pr1:forall c : R,
0 < c < x -> derivable_pt (atan - ps_atan) c
pr2:forall c : R, 0 < c < x -> derivable_pt id c
d:R
d_encad:0 < d < x
Main:(id x - id 0) *
derive_pt (atan - ps_atan) d (pr1 d d_encad) =
((atan - ps_atan)%F x - (atan - ps_atan)%F 0) *
derive_pt id d (pr2 d d_encad)
Temp:forall pr : derivable_pt (atan - ps_atan) d,
derive_pt (atan - ps_atan) d pr = 0
iatan0:atan 0 = 0
id = id
tauto.
Qed.


Theorem Alt_PI_eq : Alt_PI = PI.Alt_PI = PI
Proof.Alt_PI = PI
apply Rmult_eq_reg_r with (/4); fold (Alt_PI/4); fold (PI/4);
  [ | apply Rgt_not_eq; lra].Alt_PI / 4 = PI / 4
assert (0 < PI/6) by (apply PI6_RGT_0).H:0 < PI / 6
Alt_PI / 4 = PI / 4
assert (t1:= PI2_1).H:0 < PI / 6
t1:1 < PI / 2
Alt_PI / 4 = PI / 4
assert (t2 := PI_4).H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
Alt_PI / 4 = PI / 4
assert (m := Alt_PI_RGT_0).H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
Alt_PI / 4 = PI / 4
assert (-PI/2 < 1 < PI/2) by (rewrite Ropp_div; split; lra).H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
Alt_PI / 4 = PI / 4
apply cond_eq; intros eps ep.H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
Rabs (Alt_PI / 4 - PI / 4) < eps
change (R_dist (Alt_PI/4) (PI/4) < eps).H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
R_dist (Alt_PI / 4) (PI / 4) < eps
assert (ca : continuity_pt atan 1).H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
continuity_pt atan 1
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
R_dist (Alt_PI / 4) (PI / 4) < eps
  apply derivable_continuous_pt, derivable_pt_atan.H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
R_dist (Alt_PI / 4) (PI / 4) < eps
assert (Xe : exists eps', exists eps'',
  eps' + eps'' <= eps /\ 0 < eps' /\ 0 < eps'').H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
exists eps' eps'' : R,
  eps' + eps'' <= eps /\ 0 < eps' /\ 0 < eps''
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
Xe:exists eps' eps'' : R,
  eps' + eps'' <= eps /\ 0 < eps' /\ 0 < eps''
R_dist (Alt_PI / 4) (PI / 4) < eps
 exists (eps/2); exists (eps/2); repeat apply conj; lra.H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
Xe:exists eps' eps'' : R,
  eps' + eps'' <= eps /\ 0 < eps' /\ 0 < eps''
R_dist (Alt_PI / 4) (PI / 4) < eps
destruct Xe as [eps' [eps'' [eps_ineq [ep' ep'']]]].H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
R_dist (Alt_PI / 4) (PI / 4) < eps
destruct (ps_atan_continuity_pt_1 _ ep') as [alpha [a0 Palpha]].H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
R_dist (Alt_PI / 4) (PI / 4) < eps
destruct (ca _ ep'') as [beta [b0 Pbeta]].H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
R_dist (Alt_PI / 4) (PI / 4) < eps
assert (Xa : exists a, 0 < a < 1 /\ R_dist a 1 < alpha /\
                R_dist a 1 < beta).H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
exists a : R,
  0 < a < 1 /\ R_dist a 1 < alpha /\ R_dist a 1 < beta
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
Xa:exists a : R,
  0 < a < 1 /\
  R_dist a 1 < alpha /\ R_dist a 1 < beta
R_dist (Alt_PI / 4) (PI / 4) < eps
 exists (Rmax (/2) (Rmax (1 - alpha /2) (1 - beta /2))).H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
0 < Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2)) <
1 /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
alpha /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
beta
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
Xa:exists a : R,
  0 < a < 1 /\
  R_dist a 1 < alpha /\ R_dist a 1 < beta
R_dist (Alt_PI / 4) (PI / 4) < eps
 assert (/2 <= Rmax (/2) (Rmax (1 - alpha /2) (1 - beta /2))) by apply Rmax_l.H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
H1:/ 2 <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
0 < Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2)) <
1 /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
alpha /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
beta
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
Xa:exists a : R,
  0 < a < 1 /\
  R_dist a 1 < alpha /\ R_dist a 1 < beta
R_dist (Alt_PI / 4) (PI / 4) < eps
 assert (Rmax (1 - alpha /2) (1 - beta /2) <=
         Rmax (/2) (Rmax (1 - alpha /2) (1 - beta /2))) by apply Rmax_r.H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
H1:/ 2 <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H2:Rmax (1 - alpha / 2) (1 - beta / 2) <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
0 < Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2)) <
1 /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
alpha /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
beta
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
Xa:exists a : R,
  0 < a < 1 /\
  R_dist a 1 < alpha /\ R_dist a 1 < beta
R_dist (Alt_PI / 4) (PI / 4) < eps
 assert ((1 - alpha /2) <= Rmax (1 - alpha /2) (1 - beta /2)) by apply Rmax_l.H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
H1:/ 2 <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H2:Rmax (1 - alpha / 2) (1 - beta / 2) <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H3:1 - alpha / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
0 < Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2)) <
1 /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
alpha /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
beta
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
Xa:exists a : R,
  0 < a < 1 /\
  R_dist a 1 < alpha /\ R_dist a 1 < beta
R_dist (Alt_PI / 4) (PI / 4) < eps
 assert ((1 - beta /2) <= Rmax (1 - alpha /2) (1 - beta /2)) by apply Rmax_r.H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
H1:/ 2 <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H2:Rmax (1 - alpha / 2) (1 - beta / 2) <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H3:1 - alpha / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H4:1 - beta / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
0 < Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2)) <
1 /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
alpha /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
beta
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
Xa:exists a : R,
  0 < a < 1 /\
  R_dist a 1 < alpha /\ R_dist a 1 < beta
R_dist (Alt_PI / 4) (PI / 4) < eps
 assert (Rmax (1 - alpha /2) (1 - beta /2) < 1)
   by (apply Rmax_lub_lt; lra).H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
H1:/ 2 <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H2:Rmax (1 - alpha / 2) (1 - beta / 2) <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H3:1 - alpha / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H4:1 - beta / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H5:Rmax (1 - alpha / 2) (1 - beta / 2) < 1
0 < Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2)) <
1 /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
alpha /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
beta
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
Xa:exists a : R,
  0 < a < 1 /\
  R_dist a 1 < alpha /\ R_dist a 1 < beta
R_dist (Alt_PI / 4) (PI / 4) < eps
 split;[split;[ | apply Rmax_lub_lt]; lra | ].H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
H1:/ 2 <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H2:Rmax (1 - alpha / 2) (1 - beta / 2) <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H3:1 - alpha / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H4:1 - beta / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H5:Rmax (1 - alpha / 2) (1 - beta / 2) < 1
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
alpha /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
beta
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
Xa:exists a : R,
  0 < a < 1 /\
  R_dist a 1 < alpha /\ R_dist a 1 < beta
R_dist (Alt_PI / 4) (PI / 4) < eps
 assert (0 <= 1 - Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))).H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
H1:/ 2 <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H2:Rmax (1 - alpha / 2) (1 - beta / 2) <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H3:1 - alpha / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H4:1 - beta / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H5:Rmax (1 - alpha / 2) (1 - beta / 2) < 1
0 <=
1 - Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
H1:/ 2 <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H2:Rmax (1 - alpha / 2) (1 - beta / 2) <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H3:1 - alpha / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H4:1 - beta / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H5:Rmax (1 - alpha / 2) (1 - beta / 2) < 1
H6:0 <=
1 -
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
alpha /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
beta
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
Xa:exists a : R,
  0 < a < 1 /\
  R_dist a 1 < alpha /\ R_dist a 1 < beta
R_dist (Alt_PI / 4) (PI / 4) < eps
  assert (Rmax (/2) (Rmax (1 - alpha / 2) 
            (1 - beta /2)) <= 1) by (apply Rmax_lub; lra).H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
H1:/ 2 <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H2:Rmax (1 - alpha / 2) (1 - beta / 2) <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H3:1 - alpha / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H4:1 - beta / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H5:Rmax (1 - alpha / 2) (1 - beta / 2) < 1
H6:Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2)) <=
1
0 <=
1 - Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
H1:/ 2 <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H2:Rmax (1 - alpha / 2) (1 - beta / 2) <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H3:1 - alpha / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H4:1 - beta / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H5:Rmax (1 - alpha / 2) (1 - beta / 2) < 1
H6:0 <=
1 -
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
alpha /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
beta
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
Xa:exists a : R,
  0 < a < 1 /\
  R_dist a 1 < alpha /\ R_dist a 1 < beta
R_dist (Alt_PI / 4) (PI / 4) < eps
  lra.H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
H1:/ 2 <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H2:Rmax (1 - alpha / 2) (1 - beta / 2) <=
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
H3:1 - alpha / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H4:1 - beta / 2 <=
Rmax (1 - alpha / 2) (1 - beta / 2)
H5:Rmax (1 - alpha / 2) (1 - beta / 2) < 1
H6:0 <=
1 -
Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
alpha /\
R_dist
  (Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))) 1 <
beta
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
Xa:exists a : R,
  0 < a < 1 /\
  R_dist a 1 < alpha /\ R_dist a 1 < beta
R_dist (Alt_PI / 4) (PI / 4) < eps
 split; unfold R_dist; rewrite <-Rabs_Ropp, Ropp_minus_distr,
   Rabs_pos_eq;lra.H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
Xa:exists a : R,
  0 < a < 1 /\
  R_dist a 1 < alpha /\ R_dist a 1 < beta
R_dist (Alt_PI / 4) (PI / 4) < eps
destruct Xa as [a [[Pa0 Pa1] [P1 P2]]].H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
a:R
Pa0:0 < a
Pa1:a < 1
P1:R_dist a 1 < alpha
P2:R_dist a 1 < beta
R_dist (Alt_PI / 4) (PI / 4) < eps
apply Rle_lt_trans with (1 := R_dist_tri _ _ (ps_atan a)).H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
a:R
Pa0:0 < a
Pa1:a < 1
P1:R_dist a 1 < alpha
P2:R_dist a 1 < beta
R_dist (Alt_PI / 4) (ps_atan a) +
R_dist (ps_atan a) (PI / 4) < eps
apply Rlt_le_trans with (2 := eps_ineq).H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
a:R
Pa0:0 < a
Pa1:a < 1
P1:R_dist a 1 < alpha
P2:R_dist a 1 < beta
R_dist (Alt_PI / 4) (ps_atan a) +
R_dist (ps_atan a) (PI / 4) < 
eps' + eps''
apply Rplus_lt_compat.H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
a:R
Pa0:0 < a
Pa1:a < 1
P1:R_dist a 1 < alpha
P2:R_dist a 1 < beta
R_dist (Alt_PI / 4) (ps_atan a) < eps'
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
a:R
Pa0:0 < a
Pa1:a < 1
P1:R_dist a 1 < alpha
P2:R_dist a 1 < beta
R_dist (ps_atan a) (PI / 4) < eps''
rewrite R_dist_sym; apply Palpha; assumption.H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
a:R
Pa0:0 < a
Pa1:a < 1
P1:R_dist a 1 < alpha
P2:R_dist a 1 < beta
R_dist (ps_atan a) (PI / 4) < eps''
rewrite <- atan_eq_ps_atan.H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
a:R
Pa0:0 < a
Pa1:a < 1
P1:R_dist a 1 < alpha
P2:R_dist a 1 < beta
R_dist (atan a) (PI / 4) < eps''
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
a:R
Pa0:0 < a
Pa1:a < 1
P1:R_dist a 1 < alpha
P2:R_dist a 1 < beta
0 < a < 1
 rewrite <- atan_1; apply (Pbeta a); auto.H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
a:R
Pa0:0 < a
Pa1:a < 1
P1:R_dist a 1 < alpha
P2:R_dist a 1 < beta
D_x no_cond 1 a /\ dist R_met a 1 < beta
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
a:R
Pa0:0 < a
Pa1:a < 1
P1:R_dist a 1 < alpha
P2:R_dist a 1 < beta
0 < a < 1
 split; [ | exact P2].H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
a:R
Pa0:0 < a
Pa1:a < 1
P1:R_dist a 1 < alpha
P2:R_dist a 1 < beta
D_x no_cond 1 a
H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
a:R
Pa0:0 < a
Pa1:a < 1
P1:R_dist a 1 < alpha
P2:R_dist a 1 < beta
0 < a < 1
split;[exact I | apply Rgt_not_eq; assumption].H:0 < PI / 6
t1:1 < PI / 2
t2:PI <= 4
m:0 < Alt_PI
H0:- PI / 2 < 1 < PI / 2
eps:R
ep:0 < eps
ca:continuity_pt atan 1
eps', eps'':R
eps_ineq:eps' + eps'' <= eps
ep':0 < eps'
ep'':0 < eps''
alpha:R
a0:alpha > 0
Palpha:forall x : R,
x < 1 ->
0 < x ->
R_dist x 1 < alpha ->
dist R_met (ps_atan x) (Alt_PI / 4) < eps'
beta:R
b0:beta > 0
Pbeta:forall x : Base R_met,
D_x no_cond 1 x /\ dist R_met x 1 < beta ->
dist R_met (atan x) (atan 1) < eps''
a:R
Pa0:0 < a
Pa1:a < 1
P1:R_dist a 1 < alpha
P2:R_dist a 1 < beta
0 < a < 1
split; assumption.
Qed.

Lemma PI_ineq :
  forall N : nat,
    sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= PI / 4 <=
    sum_f_R0 (tg_alt PI_tg) (2 * N).forall N : nat,
sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= PI / 4 <=
sum_f_R0 (tg_alt PI_tg) (2 * N)
Proof.forall N : nat,
sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= PI / 4 <=
sum_f_R0 (tg_alt PI_tg) (2 * N)
intros; rewrite <- Alt_PI_eq; apply Alt_PI_ineq.
Qed.