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Diffstat (limited to 'theories/Reals/Rtrigo_alt.v')
| -rw-r--r-- | theories/Reals/Rtrigo_alt.v | 426 |
1 files changed, 426 insertions, 0 deletions
diff --git a/theories/Reals/Rtrigo_alt.v b/theories/Reals/Rtrigo_alt.v new file mode 100644 index 00000000..3cda9290 --- /dev/null +++ b/theories/Reals/Rtrigo_alt.v @@ -0,0 +1,426 @@ +(************************************************************************) +(* 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.
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