path: root/theories/Reals/Rtrigo_alt.v

(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)
 
(*i $Id: Rtrigo_alt.v,v 1.16.2.1 2004/07/16 19:31:14 herbelin Exp $ i*)

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo_def.
Open Local Scope R_scope.

(*****************************************************************)
(* Using series definitions of cos and sin                       *)
(*****************************************************************)

Definition sin_term (a:R) (i:nat) : R :=
  (-1) ^ i * (a ^ (2 * i + 1) / INR (fact (2 * i + 1))).

Definition cos_term (a:R) (i:nat) : R :=
  (-1) ^ i * (a ^ (2 * i) / INR (fact (2 * i))).

Definition sin_approx (a:R) (n:nat) : R := sum_f_R0 (sin_term a) n.

Definition cos_approx (a:R) (n:nat) : R := sum_f_R0 (cos_term a) n.

(**********)
Lemma PI_4 : PI <= 4.
assert (H0 := PI_ineq 0).
elim H0; clear H0; intros _ H0.
unfold tg_alt, PI_tg in H0; simpl in H0.
rewrite Rinv_1 in H0; rewrite Rmult_1_r in H0; unfold Rdiv in H0.
apply Rmult_le_reg_l with (/ 4).
apply Rinv_0_lt_compat; prove_sup0.
rewrite <- Rinv_l_sym; [ rewrite Rmult_comm; assumption | discrR ].
Qed.

(**********)
Theorem sin_bound :
 forall (a:R) (n:nat),
   0 <= a ->
   a <= PI -> sin_approx a (2 * n + 1) <= sin a <= sin_approx a (2 * (n + 1)).
intros; case (Req_dec a 0); intro Hyp_a.
rewrite Hyp_a; rewrite sin_0; split; right; unfold sin_approx in |- *;
 apply sum_eq_R0 || (symmetry  in |- *; apply sum_eq_R0); 
 intros; unfold sin_term in |- *; rewrite pow_add; 
 simpl in |- *; unfold Rdiv in |- *; rewrite Rmult_0_l; 
 ring.
unfold sin_approx in |- *; cut (0 < a).
intro Hyp_a_pos.
rewrite (decomp_sum (sin_term a) (2 * n + 1)).
rewrite (decomp_sum (sin_term a) (2 * (n + 1))).
replace (sin_term a 0) with a.
cut
 (sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * n + 1)) <= sin a - a /\
  sin a - a <= sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * (n + 1))) ->
  a + sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * n + 1)) <= sin a /\
  sin a <= a + sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * (n + 1)))).
intro; apply H1.
set (Un := fun n:nat => a ^ (2 * S n + 1) / INR (fact (2 * S n + 1))).
replace (pred (2 * n + 1)) with (2 * n)%nat.
replace (pred (2 * (n + 1))) with (S (2 * n)).
replace (sum_f_R0 (fun i:nat => sin_term a (S i)) (2 * n)) with
 (- sum_f_R0 (tg_alt Un) (2 * n)).
replace (sum_f_R0 (fun i:nat => sin_term a (S i)) (S (2 * n))) with
 (- sum_f_R0 (tg_alt Un) (S (2 * n))).
cut
 (sum_f_R0 (tg_alt Un) (S (2 * n)) <= a - sin a <=
  sum_f_R0 (tg_alt Un) (2 * n) ->
  - sum_f_R0 (tg_alt Un) (2 * n) <= sin a - a <=
  - sum_f_R0 (tg_alt Un) (S (2 * n))).
intro; apply H2.
apply alternated_series_ineq.
unfold Un_decreasing, Un in |- *; intro;
 cut ((2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))).
intro; rewrite H3.
replace (a ^ S (S (2 * S n0 + 1))) with (a ^ (2 * S n0 + 1) * (a * a)).
unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l.
left; apply pow_lt; assumption.
apply Rmult_le_reg_l with (INR (fact (S (S (2 * S n0 + 1))))).
rewrite <- H3; apply lt_INR_0; apply neq_O_lt; red in |- *; intro;
 assert (H5 := sym_eq H4); elim (fact_neq_0 _ H5).
rewrite <- H3; rewrite (Rmult_comm (INR (fact (2 * S (S n0) + 1))));
 rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; rewrite H3; do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
 repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
rewrite Rmult_1_r.
do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR;
 simpl in |- *;
 replace
  (((0 + 1 + 1) * (INR n0 + 1) + (0 + 1) + 1 + 1) *
   ((0 + 1 + 1) * (INR n0 + 1) + (0 + 1) + 1)) with
  (4 * INR n0 * INR n0 + 18 * INR n0 + 20); [ idtac | ring ].
apply Rle_trans with 20.
apply Rle_trans with 16.
replace 16 with (Rsqr 4); [ idtac | ring_Rsqr ].
replace (a * a) with (Rsqr a); [ idtac | reflexivity ].
apply Rsqr_incr_1.
apply Rle_trans with PI; [ assumption | apply PI_4 ].
assumption.
left; prove_sup0.
rewrite <- (Rplus_0_r 16); replace 20 with (16 + 4);
 [ apply Rplus_le_compat_l; left; prove_sup0 | ring ].
rewrite <- (Rplus_comm 20); pattern 20 at 1 in |- *; rewrite <- Rplus_0_r;
 apply Rplus_le_compat_l.
apply Rplus_le_le_0_compat.
repeat apply Rmult_le_pos.
left; prove_sup0.
left; prove_sup0.
replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ].
replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ].
apply Rmult_le_pos.
left; prove_sup0.
replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ].
apply INR_fact_neq_0.
apply INR_fact_neq_0.
simpl in |- *; ring.
apply INR_eq; do 2 rewrite S_INR; do 2 rewrite plus_INR;
 do 2 rewrite mult_INR; repeat rewrite S_INR; ring.
assert (H3 := cv_speed_pow_fact a); unfold Un in |- *; unfold Un_cv in H3;
 unfold R_dist in H3; unfold Un_cv in |- *; unfold R_dist in |- *; 
 intros; elim (H3 eps H4); intros N H5.
exists N; intros; apply H5.
replace (2 * S n0 + 1)%nat with (S (2 * S n0)).
unfold ge in |- *; apply le_trans with (2 * S n0)%nat.
apply le_trans with (2 * S N)%nat.
apply le_trans with (2 * N)%nat.
apply le_n_2n.
apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_Sn.
apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_S; assumption.
apply le_n_Sn.
apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR; reflexivity.
assert (X := exist_sin (Rsqr a)); elim X; intros.
cut (x = sin a / a).
intro; rewrite H3 in p; unfold sin_in in p; unfold infinit_sum in p;
 unfold R_dist in p; unfold Un_cv in |- *; unfold R_dist in |- *; 
 intros.
cut (0 < eps / Rabs a).
intro; elim (p _ H5); intros N H6.
exists N; intros.
replace (sum_f_R0 (tg_alt Un) n0) with
 (a * (1 - sum_f_R0 (fun i:nat => sin_n i * Rsqr a ^ i) (S n0))).
unfold Rminus in |- *; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r;
 rewrite Ropp_plus_distr; rewrite Ropp_involutive; 
 repeat rewrite Rplus_assoc; rewrite (Rplus_comm a);
 rewrite (Rplus_comm (- a)); repeat rewrite Rplus_assoc; 
 rewrite Rplus_opp_l; rewrite Rplus_0_r; apply Rmult_lt_reg_l with (/ Rabs a).
apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
pattern (/ Rabs a) at 1 in |- *; rewrite <- (Rabs_Rinv a Hyp_a).
rewrite <- Rabs_mult; rewrite Rmult_plus_distr_l; rewrite <- Rmult_assoc;
 rewrite <- Rinv_l_sym; [ rewrite Rmult_1_l | assumption ];
 rewrite (Rmult_comm (/ a)); rewrite (Rmult_comm (/ Rabs a));
 rewrite <- Rabs_Ropp; rewrite Ropp_plus_distr; rewrite Ropp_involutive;
 unfold Rminus, Rdiv in H6; apply H6; unfold ge in |- *;
 apply le_trans with n0; [ exact H7 | apply le_n_Sn ].
rewrite (decomp_sum (fun i:nat => sin_n i * Rsqr a ^ i) (S n0)).
replace (sin_n 0) with 1.
simpl in |- *; rewrite Rmult_1_r; unfold Rminus in |- *;
 rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r;
 rewrite Rplus_0_l; rewrite Ropp_mult_distr_r_reverse;
 rewrite <- Ropp_mult_distr_l_reverse; rewrite scal_sum; 
 apply sum_eq.
intros; unfold sin_n, Un, tg_alt in |- *;
 replace ((-1) ^ S i) with (- (-1) ^ i).
replace (a ^ (2 * S i + 1)) with (Rsqr a * Rsqr a ^ i * a).
unfold Rdiv in |- *; ring.
rewrite pow_add; rewrite pow_Rsqr; simpl in |- *; ring.
simpl in |- *; ring.
unfold sin_n in |- *; unfold Rdiv in |- *; simpl in |- *; rewrite Rinv_1;
 rewrite Rmult_1_r; reflexivity.
apply lt_O_Sn.
unfold Rdiv in |- *; apply Rmult_lt_0_compat.
assumption.
apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
unfold sin in |- *; case (exist_sin (Rsqr a)).
intros; cut (x = x0).
intro; rewrite H3; unfold Rdiv in |- *.
symmetry  in |- *; apply Rinv_r_simpl_m; assumption.
unfold sin_in in p; unfold sin_in in s; eapply uniqueness_sum.
apply p.
apply s.
intros; elim H2; intros.
replace (sin a - a) with (- (a - sin a)); [ idtac | ring ].
split; apply Ropp_le_contravar; assumption.
replace (- sum_f_R0 (tg_alt Un) (S (2 * n))) with
 (-1 * sum_f_R0 (tg_alt Un) (S (2 * n))); [ rewrite scal_sum | ring ].
apply sum_eq; intros; unfold sin_term, Un, tg_alt in |- *;
 replace ((-1) ^ S i) with (-1 * (-1) ^ i).
unfold Rdiv in |- *; ring.
reflexivity.
replace (- sum_f_R0 (tg_alt Un) (2 * n)) with
 (-1 * sum_f_R0 (tg_alt Un) (2 * n)); [ rewrite scal_sum | ring ].
apply sum_eq; intros.
unfold sin_term, Un, tg_alt in |- *;
 replace ((-1) ^ S i) with (-1 * (-1) ^ i).
unfold Rdiv in |- *; ring.
reflexivity.
replace (2 * (n + 1))%nat with (S (S (2 * n))).
reflexivity.
apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; rewrite plus_INR;
 repeat rewrite S_INR; ring.
replace (2 * n + 1)%nat with (S (2 * n)).
reflexivity.
apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR;
 repeat rewrite S_INR; ring.
intro; elim H1; intros.
split.
apply Rplus_le_reg_l with (- a).
rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
 rewrite (Rplus_comm (- a)); apply H2.
apply Rplus_le_reg_l with (- a).
rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
 rewrite (Rplus_comm (- a)); apply H3.
unfold sin_term in |- *; simpl in |- *; unfold Rdiv in |- *; rewrite Rinv_1;
 ring.
replace (2 * (n + 1))%nat with (S (S (2 * n))).
apply lt_O_Sn.
apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; rewrite plus_INR;
 repeat rewrite S_INR; ring.
replace (2 * n + 1)%nat with (S (2 * n)).
apply lt_O_Sn.
apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR;
 repeat rewrite S_INR; ring.
inversion H; [ assumption | elim Hyp_a; symmetry  in |- *; assumption ].
Qed.

(**********)
Lemma cos_bound :
 forall (a:R) (n:nat),
   - PI / 2 <= a ->
   a <= PI / 2 ->
   cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1)). 
cut
 ((forall (a:R) (n:nat),
     0 <= a ->
     a <= PI / 2 ->
     cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1))) ->
  forall (a:R) (n:nat),
    - PI / 2 <= a ->
    a <= PI / 2 ->
    cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1))).
intros H a n; apply H.
intros; unfold cos_approx in |- *.
rewrite (decomp_sum (cos_term a0) (2 * n0 + 1)).
rewrite (decomp_sum (cos_term a0) (2 * (n0 + 1))).
replace (cos_term a0 0) with 1.
cut
 (sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * n0 + 1)) <= cos a0 - 1 /\
  cos a0 - 1 <=
  sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * (n0 + 1))) ->
  1 + sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * n0 + 1)) <= cos a0 /\
  cos a0 <=
  1 + sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * (n0 + 1)))).
intro; apply H2.
set (Un := fun n:nat => a0 ^ (2 * S n) / INR (fact (2 * S n))).
replace (pred (2 * n0 + 1)) with (2 * n0)%nat.
replace (pred (2 * (n0 + 1))) with (S (2 * n0)).
replace (sum_f_R0 (fun i:nat => cos_term a0 (S i)) (2 * n0)) with
 (- sum_f_R0 (tg_alt Un) (2 * n0)).
replace (sum_f_R0 (fun i:nat => cos_term a0 (S i)) (S (2 * n0))) with
 (- sum_f_R0 (tg_alt Un) (S (2 * n0))).
cut
 (sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
  sum_f_R0 (tg_alt Un) (2 * n0) ->
  - sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
  - sum_f_R0 (tg_alt Un) (S (2 * n0))).
intro; apply H3.
apply alternated_series_ineq.
unfold Un_decreasing in |- *; intro; unfold Un in |- *.
cut ((2 * S (S n1))%nat = S (S (2 * S n1))).
intro; rewrite H4;
 replace (a0 ^ S (S (2 * S n1))) with (a0 ^ (2 * S n1) * (a0 * a0)).
unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l.
apply pow_le; assumption.
apply Rmult_le_reg_l with (INR (fact (S (S (2 * S n1))))).
rewrite <- H4; apply lt_INR_0; apply neq_O_lt; red in |- *; intro;
 assert (H6 := sym_eq H5); elim (fact_neq_0 _ H6).
rewrite <- H4; rewrite (Rmult_comm (INR (fact (2 * S (S n1)))));
 rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; rewrite H4; do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
 repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
rewrite Rmult_1_r; do 2 rewrite S_INR; rewrite mult_INR; repeat rewrite S_INR;
 simpl in |- *;
 replace
  (((0 + 1 + 1) * (INR n1 + 1) + 1 + 1) * ((0 + 1 + 1) * (INR n1 + 1) + 1))
  with (4 * INR n1 * INR n1 + 14 * INR n1 + 12); [ idtac | ring ].
apply Rle_trans with 12.
apply Rle_trans with 4.
replace 4 with (Rsqr 2); [ idtac | ring_Rsqr ].
replace (a0 * a0) with (Rsqr a0); [ idtac | reflexivity ].
apply Rsqr_incr_1.
apply Rle_trans with (PI / 2).
assumption.
unfold Rdiv in |- *; apply Rmult_le_reg_l with 2.
prove_sup0.
rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m.
replace 4 with 4; [ apply PI_4 | ring ].
discrR.
assumption.
left; prove_sup0.
pattern 4 at 1 in |- *; rewrite <- Rplus_0_r; replace 12 with (4 + 8);
 [ apply Rplus_le_compat_l; left; prove_sup0 | ring ].
rewrite <- (Rplus_comm 12); pattern 12 at 1 in |- *; rewrite <- Rplus_0_r;
 apply Rplus_le_compat_l.
apply Rplus_le_le_0_compat.
repeat apply Rmult_le_pos.
left; prove_sup0.
left; prove_sup0.
replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ].
replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ].
apply Rmult_le_pos.
left; prove_sup0.
replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ].
apply INR_fact_neq_0.
apply INR_fact_neq_0.
simpl in |- *; ring.
apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR;
 ring.
assert (H4 := cv_speed_pow_fact a0); unfold Un in |- *; unfold Un_cv in H4;
 unfold R_dist in H4; unfold Un_cv in |- *; unfold R_dist in |- *; 
 intros; elim (H4 eps H5); intros N H6; exists N; intros.
apply H6; unfold ge in |- *; apply le_trans with (2 * S N)%nat.
apply le_trans with (2 * N)%nat.
apply le_n_2n.
apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_Sn.
apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_S; assumption.
assert (X := exist_cos (Rsqr a0)); elim X; intros.
cut (x = cos a0).
intro; rewrite H4 in p; unfold cos_in in p; unfold infinit_sum in p;
 unfold R_dist in p; unfold Un_cv in |- *; unfold R_dist in |- *; 
 intros.
elim (p _ H5); intros N H6.
exists N; intros.
replace (sum_f_R0 (tg_alt Un) n1) with
 (1 - sum_f_R0 (fun i:nat => cos_n i * Rsqr a0 ^ i) (S n1)).
unfold Rminus in |- *; rewrite Ropp_plus_distr; rewrite Ropp_involutive;
 repeat rewrite Rplus_assoc; rewrite (Rplus_comm 1);
 rewrite (Rplus_comm (-1)); repeat rewrite Rplus_assoc; 
 rewrite Rplus_opp_l; rewrite Rplus_0_r; rewrite <- Rabs_Ropp;
 rewrite Ropp_plus_distr; rewrite Ropp_involutive; 
 unfold Rminus in H6; apply H6.
unfold ge in |- *; apply le_trans with n1.
exact H7.
apply le_n_Sn.
rewrite (decomp_sum (fun i:nat => cos_n i * Rsqr a0 ^ i) (S n1)).
replace (cos_n 0) with 1.
simpl in |- *; rewrite Rmult_1_r; unfold Rminus in |- *;
 rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r;
 rewrite Rplus_0_l;
 replace (- sum_f_R0 (fun i:nat => cos_n (S i) * (Rsqr a0 * Rsqr a0 ^ i)) n1)
  with
  (-1 * sum_f_R0 (fun i:nat => cos_n (S i) * (Rsqr a0 * Rsqr a0 ^ i)) n1);
 [ idtac | ring ]; rewrite scal_sum; apply sum_eq; 
 intros; unfold cos_n, Un, tg_alt in |- *.
replace ((-1) ^ S i) with (- (-1) ^ i).
replace (a0 ^ (2 * S i)) with (Rsqr a0 * Rsqr a0 ^ i).
unfold Rdiv in |- *; ring.
rewrite pow_Rsqr; reflexivity.
simpl in |- *; ring.
unfold cos_n in |- *; unfold Rdiv in |- *; simpl in |- *; rewrite Rinv_1;
 rewrite Rmult_1_r; reflexivity.
apply lt_O_Sn.
unfold cos in |- *; case (exist_cos (Rsqr a0)); intros; unfold cos_in in p;
 unfold cos_in in c; eapply uniqueness_sum.
apply p.
apply c.
intros; elim H3; intros; replace (cos a0 - 1) with (- (1 - cos a0));
 [ idtac | ring ].
split; apply Ropp_le_contravar; assumption.
replace (- sum_f_R0 (tg_alt Un) (S (2 * n0))) with
 (-1 * sum_f_R0 (tg_alt Un) (S (2 * n0))); [ rewrite scal_sum | ring ].
apply sum_eq; intros; unfold cos_term, Un, tg_alt in |- *;
 replace ((-1) ^ S i) with (-1 * (-1) ^ i).
unfold Rdiv in |- *; ring.
reflexivity.
replace (- sum_f_R0 (tg_alt Un) (2 * n0)) with
 (-1 * sum_f_R0 (tg_alt Un) (2 * n0)); [ rewrite scal_sum | ring ];
 apply sum_eq; intros; unfold cos_term, Un, tg_alt in |- *;
 replace ((-1) ^ S i) with (-1 * (-1) ^ i).
unfold Rdiv in |- *; ring.
reflexivity.
replace (2 * (n0 + 1))%nat with (S (S (2 * n0))).
reflexivity.
apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; rewrite plus_INR;
 repeat rewrite S_INR; ring.
replace (2 * n0 + 1)%nat with (S (2 * n0)).
reflexivity.
apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR;
 repeat rewrite S_INR; ring.
intro; elim H2; intros; split.
apply Rplus_le_reg_l with (-1).
rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
 rewrite (Rplus_comm (-1)); apply H3.
apply Rplus_le_reg_l with (-1).
rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
 rewrite (Rplus_comm (-1)); apply H4.
unfold cos_term in |- *; simpl in |- *; unfold Rdiv in |- *; rewrite Rinv_1;
 ring.
replace (2 * (n0 + 1))%nat with (S (S (2 * n0))).
apply lt_O_Sn.
apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; rewrite plus_INR;
 repeat rewrite S_INR; ring.
replace (2 * n0 + 1)%nat with (S (2 * n0)).
apply lt_O_Sn.
apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR;
 repeat rewrite S_INR; ring.
intros; case (total_order_T 0 a); intro.
elim s; intro.
apply H; [ left; assumption | assumption ].
apply H; [ right; assumption | assumption ].
cut (0 < - a).
intro; cut (forall (x:R) (n:nat), cos_approx x n = cos_approx (- x) n).
intro; rewrite H3; rewrite (H3 a (2 * (n + 1))%nat); rewrite cos_sym; apply H.
left; assumption.
rewrite <- (Ropp_involutive (PI / 2)); apply Ropp_le_contravar;
 unfold Rdiv in |- *; unfold Rdiv in H0; rewrite <- Ropp_mult_distr_l_reverse;
 exact H0.
intros; unfold cos_approx in |- *; apply sum_eq; intros;
 unfold cos_term in |- *; do 2 rewrite pow_Rsqr; rewrite Rsqr_neg;
 unfold Rdiv in |- *; reflexivity.
apply Ropp_0_gt_lt_contravar; assumption.
Qed.