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Diffstat (limited to 'theories/Reals/Rtrigo_reg.v')
| -rw-r--r-- | theories/Reals/Rtrigo_reg.v | 608 |
1 files changed, 608 insertions, 0 deletions
diff --git a/theories/Reals/Rtrigo_reg.v b/theories/Reals/Rtrigo_reg.v new file mode 100644 index 00000000..9d3b60c6 --- /dev/null +++ b/theories/Reals/Rtrigo_reg.v @@ -0,0 +1,608 @@ +(************************************************************************) +(* 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_reg.v,v 1.15.2.1 2004/07/16 19:31:15 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo. +Require Import Ranalysis1. +Require Import PSeries_reg. +Open Local Scope nat_scope. +Open Local Scope R_scope. + +Lemma CVN_R_cos : + forall fn:nat -> R -> R, + fn = (fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)) -> + CVN_R fn. +unfold CVN_R in |- *; intros. +cut ((r:R) <> 0). +intro hyp_r; unfold CVN_r in |- *. +apply existT with (fun n:nat => / INR (fact (2 * n)) * r ^ (2 * n)). +cut + (sigT + (fun l:R => + Un_cv + (fun n:nat => + sum_f_R0 (fun k:nat => Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) + n) l)). +intro; elim X; intros. +apply existT with x. +split. +apply p. +intros; rewrite H; unfold Rdiv in |- *; do 2 rewrite Rabs_mult. +rewrite pow_1_abs; rewrite Rmult_1_l. +cut (0 < / INR (fact (2 * n))). +intro; rewrite (Rabs_right _ (Rle_ge _ _ (Rlt_le _ _ H1))). +apply Rmult_le_compat_l. +left; apply H1. +rewrite <- RPow_abs; apply pow_maj_Rabs. +rewrite Rabs_Rabsolu. +unfold Boule in H0; rewrite Rminus_0_r in H0. +left; apply H0. +apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Alembert_C2. +intro; apply Rabs_no_R0. +apply prod_neq_R0. +apply Rinv_neq_0_compat. +apply INR_fact_neq_0. +apply pow_nonzero; assumption. +assert (H0 := Alembert_cos). +unfold cos_n in H0; unfold Un_cv in H0; unfold Un_cv in |- *; intros. +cut (0 < eps / Rsqr r). +intro; elim (H0 _ H2); intros N0 H3. +exists N0; intros. +unfold R_dist in |- *; assert (H5 := H3 _ H4). +unfold R_dist in H5; + replace + (Rabs + (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) / + Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) with + (Rsqr r * + Rabs ((-1) ^ S n / INR (fact (2 * S n)) / ((-1) ^ n / INR (fact (2 * n))))). +apply Rmult_lt_reg_l with (/ Rsqr r). +apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption. +pattern (/ Rsqr r) at 1 in |- *; replace (/ Rsqr r) with (Rabs (/ Rsqr r)). +rewrite <- Rabs_mult; rewrite Rmult_minus_distr_l; rewrite Rmult_0_r; + rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); apply H5. +unfold Rsqr in |- *; apply prod_neq_R0; assumption. +rewrite Rabs_Rinv. +rewrite Rabs_right. +reflexivity. +apply Rle_ge; apply Rle_0_sqr. +unfold Rsqr in |- *; apply prod_neq_R0; assumption. +rewrite (Rmult_comm (Rsqr r)); unfold Rdiv in |- *; repeat rewrite Rabs_mult; + rewrite Rabs_Rabsolu; rewrite pow_1_abs; rewrite Rmult_1_l; + repeat rewrite Rmult_assoc; apply Rmult_eq_compat_l. +rewrite Rabs_Rinv. +rewrite Rabs_mult; rewrite (pow_1_abs n); rewrite Rmult_1_l; + rewrite <- Rabs_Rinv. +rewrite Rinv_involutive. +rewrite Rinv_mult_distr. +rewrite Rabs_Rinv. +rewrite Rinv_involutive. +rewrite (Rmult_comm (Rabs (Rabs (r ^ (2 * S n))))); rewrite Rabs_mult; + rewrite Rabs_Rabsolu; rewrite Rmult_assoc; apply Rmult_eq_compat_l. +rewrite Rabs_Rinv. +do 2 rewrite Rabs_Rabsolu; repeat rewrite Rabs_right. +replace (r ^ (2 * S n)) with (r ^ (2 * n) * r * r). +repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +unfold Rsqr in |- *; ring. +apply pow_nonzero; assumption. +replace (2 * S n)%nat with (S (S (2 * n))). +simpl in |- *; ring. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +apply Rle_ge; apply pow_le; left; apply (cond_pos r). +apply Rle_ge; apply pow_le; left; apply (cond_pos r). +apply Rabs_no_R0; apply pow_nonzero; assumption. +apply Rabs_no_R0; apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply Rabs_no_R0; apply Rinv_neq_0_compat; apply INR_fact_neq_0. +apply Rabs_no_R0; apply pow_nonzero; assumption. +apply INR_fact_neq_0. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. +apply prod_neq_R0. +apply pow_nonzero; discrR. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply H1. +apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption. +assert (H0 := cond_pos r); red in |- *; intro; rewrite H1 in H0; + elim (Rlt_irrefl _ H0). +Qed. + +(**********) +Lemma continuity_cos : continuity cos. +set (fn := fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)). +cut (CVN_R fn). +intro; cut (forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)). +intro cv; cut (forall n:nat, continuity (fn n)). +intro; cut (forall x:R, cos x = SFL fn cv x). +intro; cut (continuity (SFL fn cv) -> continuity cos). +intro; apply H1. +apply SFL_continuity; assumption. +unfold continuity in |- *; unfold continuity_pt in |- *; + unfold continue_in in |- *; unfold limit1_in in |- *; + unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *; + intros. +elim (H1 x _ H2); intros. +exists x0; intros. +elim H3; intros. +split. +apply H4. +intros; rewrite (H0 x); rewrite (H0 x1); apply H5; apply H6. +intro; unfold cos, SFL in |- *. +case (cv x); case (exist_cos (Rsqr x)); intros. +symmetry in |- *; eapply UL_sequence. +apply u. +unfold cos_in in c; unfold infinit_sum in c; unfold Un_cv in |- *; intros. +elim (c _ H0); intros N0 H1. +exists N0; intros. +unfold R_dist in H1; unfold R_dist, SP in |- *. +replace (sum_f_R0 (fun k:nat => fn k x) n) with + (sum_f_R0 (fun i:nat => cos_n i * Rsqr x ^ i) n). +apply H1; assumption. +apply sum_eq; intros. +unfold cos_n, fn in |- *; apply Rmult_eq_compat_l. +unfold Rsqr in |- *; rewrite pow_sqr; reflexivity. +intro; unfold fn in |- *; + replace (fun x:R => (-1) ^ n / INR (fact (2 * n)) * x ^ (2 * n)) with + (fct_cte ((-1) ^ n / INR (fact (2 * n))) * pow_fct (2 * n))%F; + [ idtac | reflexivity ]. +apply continuity_mult. +apply derivable_continuous; apply derivable_const. +apply derivable_continuous; apply (derivable_pow (2 * n)). +apply CVN_R_CVS; apply X. +apply CVN_R_cos; unfold fn in |- *; reflexivity. +Qed. + +(**********) +Lemma continuity_sin : continuity sin. +unfold continuity in |- *; intro. +assert (H0 := continuity_cos (PI / 2 - x)). +unfold continuity_pt in H0; unfold continue_in in H0; unfold limit1_in in H0; + unfold limit_in in H0; simpl in H0; unfold R_dist in H0; + unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +elim (H0 _ H); intros. +exists x0; intros. +elim H1; intros. +split. +assumption. +intros; rewrite <- (cos_shift x); rewrite <- (cos_shift x1); apply H3. +elim H4; intros. +split. +unfold D_x, no_cond in |- *; split. +trivial. +red in |- *; intro; unfold D_x, no_cond in H5; elim H5; intros _ H8; elim H8; + rewrite <- (Ropp_involutive x); rewrite <- (Ropp_involutive x1); + apply Ropp_eq_compat; apply Rplus_eq_reg_l with (PI / 2); + apply H7. +replace (PI / 2 - x1 - (PI / 2 - x)) with (x - x1); [ idtac | ring ]; + rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply H6. +Qed. + +Lemma CVN_R_sin : + forall fn:nat -> R -> R, + fn = + (fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)) -> + CVN_R fn. +unfold CVN_R in |- *; unfold CVN_r in |- *; intros fn H r. +apply existT with (fun n:nat => / INR (fact (2 * n + 1)) * r ^ (2 * n)). +cut + (sigT + (fun l:R => + Un_cv + (fun n:nat => + sum_f_R0 + (fun k:nat => Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n) + l)). +intro; elim X; intros. +apply existT with x. +split. +apply p. +intros; rewrite H; unfold Rdiv in |- *; do 2 rewrite Rabs_mult; + rewrite pow_1_abs; rewrite Rmult_1_l. +cut (0 < / INR (fact (2 * n + 1))). +intro; rewrite (Rabs_right _ (Rle_ge _ _ (Rlt_le _ _ H1))). +apply Rmult_le_compat_l. +left; apply H1. +rewrite <- RPow_abs; apply pow_maj_Rabs. +rewrite Rabs_Rabsolu; unfold Boule in H0; rewrite Rminus_0_r in H0; left; + apply H0. +apply Rinv_0_lt_compat; apply INR_fact_lt_0. +cut ((r:R) <> 0). +intro; apply Alembert_C2. +intro; apply Rabs_no_R0. +apply prod_neq_R0. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. +apply pow_nonzero; assumption. +assert (H1 := Alembert_sin). +unfold sin_n in H1; unfold Un_cv in H1; unfold Un_cv in |- *; intros. +cut (0 < eps / Rsqr r). +intro; elim (H1 _ H3); intros N0 H4. +exists N0; intros. +unfold R_dist in |- *; assert (H6 := H4 _ H5). +unfold R_dist in H5; + replace + (Rabs + (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) / + Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))) with + (Rsqr r * + Rabs + ((-1) ^ S n / INR (fact (2 * S n + 1)) / + ((-1) ^ n / INR (fact (2 * n + 1))))). +apply Rmult_lt_reg_l with (/ Rsqr r). +apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption. +pattern (/ Rsqr r) at 1 in |- *; rewrite <- (Rabs_right (/ Rsqr r)). +rewrite <- Rabs_mult. +rewrite Rmult_minus_distr_l. +rewrite Rmult_0_r; rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite <- (Rmult_comm eps). +apply H6. +unfold Rsqr in |- *; apply prod_neq_R0; assumption. +apply Rle_ge; left; apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption. +unfold Rdiv in |- *; rewrite (Rmult_comm (Rsqr r)); repeat rewrite Rabs_mult; + rewrite Rabs_Rabsolu; rewrite pow_1_abs. +rewrite Rmult_1_l. +repeat rewrite Rmult_assoc; apply Rmult_eq_compat_l. +rewrite Rinv_mult_distr. +rewrite Rinv_involutive. +rewrite Rabs_mult. +rewrite Rabs_Rinv. +rewrite pow_1_abs; rewrite Rinv_1; rewrite Rmult_1_l. +rewrite Rinv_mult_distr. +rewrite <- Rabs_Rinv. +rewrite Rinv_involutive. +rewrite Rabs_mult. +do 2 rewrite Rabs_Rabsolu. +rewrite (Rmult_comm (Rabs (r ^ (2 * S n)))). +rewrite Rmult_assoc; apply Rmult_eq_compat_l. +rewrite Rabs_Rinv. +rewrite Rabs_Rabsolu. +repeat rewrite Rabs_right. +replace (r ^ (2 * S n)) with (r ^ (2 * n) * r * r). +do 2 rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +unfold Rsqr in |- *; ring. +apply pow_nonzero; assumption. +replace (2 * S n)%nat with (S (S (2 * n))). +simpl in |- *; ring. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +apply Rle_ge; apply pow_le; left; apply (cond_pos r). +apply Rle_ge; apply pow_le; left; apply (cond_pos r). +apply Rabs_no_R0; apply pow_nonzero; assumption. +apply INR_fact_neq_0. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. +apply Rabs_no_R0; apply Rinv_neq_0_compat; apply INR_fact_neq_0. +apply Rabs_no_R0; apply pow_nonzero; assumption. +apply pow_nonzero; discrR. +apply INR_fact_neq_0. +apply pow_nonzero; discrR. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption ]. +assert (H0 := cond_pos r); red in |- *; intro; rewrite H1 in H0; + elim (Rlt_irrefl _ H0). +Qed. + +(* (sin h)/h -> 1 when h -> 0 *) +Lemma derivable_pt_lim_sin_0 : derivable_pt_lim sin 0 1. +unfold derivable_pt_lim in |- *; intros. +set + (fn := fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)). +cut (CVN_R fn). +intro; cut (forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)). +intro cv. +set (r := mkposreal _ Rlt_0_1). +cut (CVN_r fn r). +intro; cut (forall (n:nat) (y:R), Boule 0 r y -> continuity_pt (fn n) y). +intro; cut (Boule 0 r 0). +intro; assert (H2 := SFL_continuity_pt _ cv _ X0 H0 _ H1). +unfold continuity_pt in H2; unfold continue_in in H2; unfold limit1_in in H2; + unfold limit_in in H2; simpl in H2; unfold R_dist in H2. +elim (H2 _ H); intros alp H3. +elim H3; intros. +exists (mkposreal _ H4). +simpl in |- *; intros. +rewrite sin_0; rewrite Rplus_0_l; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r. +cut (Rabs (SFL fn cv h - SFL fn cv 0) < eps). +intro; cut (SFL fn cv 0 = 1). +intro; cut (SFL fn cv h = sin h / h). +intro; rewrite H9 in H8; rewrite H10 in H8. +apply H8. +unfold SFL, sin in |- *. +case (cv h); intros. +case (exist_sin (Rsqr h)); intros. +unfold Rdiv in |- *; rewrite (Rinv_r_simpl_m h x0 H6). +eapply UL_sequence. +apply u. +unfold sin_in in s; unfold sin_n, infinit_sum in s; + unfold SP, fn, Un_cv in |- *; intros. +elim (s _ H10); intros N0 H11. +exists N0; intros. +unfold R_dist in |- *; unfold R_dist in H11. +replace + (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * h ^ (2 * k)) n) + with + (sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i + 1)) * Rsqr h ^ i) n). +apply H11; assumption. +apply sum_eq; intros; apply Rmult_eq_compat_l; unfold Rsqr in |- *; + rewrite pow_sqr; reflexivity. +unfold SFL, sin in |- *. +case (cv 0); intros. +eapply UL_sequence. +apply u. +unfold SP, fn in |- *; unfold Un_cv in |- *; intros; exists 1%nat; intros. +unfold R_dist in |- *; + replace + (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * 0 ^ (2 * k)) n) + with 1. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. +rewrite decomp_sum. +simpl in |- *; rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite Rinv_1; + rewrite Rmult_1_r; pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_eq_compat_l. +symmetry in |- *; apply sum_eq_R0; intros. +rewrite Rmult_0_l; rewrite Rmult_0_r; reflexivity. +unfold ge in H10; apply lt_le_trans with 1%nat; [ apply lt_n_Sn | apply H10 ]. +apply H5. +split. +unfold D_x, no_cond in |- *; split. +trivial. +apply (sym_not_eq (A:=R)); apply H6. +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply H7. +unfold Boule in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; rewrite Rabs_R0; apply (cond_pos r). +intros; unfold fn in |- *; + replace (fun x:R => (-1) ^ n / INR (fact (2 * n + 1)) * x ^ (2 * n)) with + (fct_cte ((-1) ^ n / INR (fact (2 * n + 1))) * pow_fct (2 * n))%F; + [ idtac | reflexivity ]. +apply continuity_pt_mult. +apply derivable_continuous_pt. +apply derivable_pt_const. +apply derivable_continuous_pt. +apply (derivable_pt_pow (2 * n) y). +apply (X r). +apply (CVN_R_CVS _ X). +apply CVN_R_sin; unfold fn in |- *; reflexivity. +Qed. + +(* ((cos h)-1)/h -> 0 when h -> 0 *) +Lemma derivable_pt_lim_cos_0 : derivable_pt_lim cos 0 0. +unfold derivable_pt_lim in |- *; intros. +assert (H0 := derivable_pt_lim_sin_0). +unfold derivable_pt_lim in H0. +cut (0 < eps / 2). +intro; elim (H0 _ H1); intros del H2. +cut (continuity_pt sin 0). +intro; unfold continuity_pt in H3; unfold continue_in in H3; + unfold limit1_in in H3; unfold limit_in in H3; simpl in H3; + unfold R_dist in H3. +cut (0 < eps / 2); [ intro | assumption ]. +elim (H3 _ H4); intros del_c H5. +cut (0 < Rmin del del_c). +intro; set (delta := mkposreal _ H6). +exists delta; intros. +rewrite Rplus_0_l; replace (cos h - cos 0) with (-2 * Rsqr (sin (h / 2))). +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r. +unfold Rdiv in |- *; do 2 rewrite Ropp_mult_distr_l_reverse. +rewrite Rabs_Ropp. +replace (2 * Rsqr (sin (h * / 2)) * / h) with + (sin (h / 2) * (sin (h / 2) / (h / 2) - 1) + sin (h / 2)). +apply Rle_lt_trans with + (Rabs (sin (h / 2) * (sin (h / 2) / (h / 2) - 1)) + Rabs (sin (h / 2))). +apply Rabs_triang. +rewrite (double_var eps); apply Rplus_lt_compat. +apply Rle_lt_trans with (Rabs (sin (h / 2) / (h / 2) - 1)). +rewrite Rabs_mult; rewrite Rmult_comm; + pattern (Rabs (sin (h / 2) / (h / 2) - 1)) at 2 in |- *; + rewrite <- Rmult_1_r; apply Rmult_le_compat_l. +apply Rabs_pos. +assert (H9 := SIN_bound (h / 2)). +unfold Rabs in |- *; case (Rcase_abs (sin (h / 2))); intro. +pattern 1 at 3 in |- *; rewrite <- (Ropp_involutive 1). +apply Ropp_le_contravar. +elim H9; intros; assumption. +elim H9; intros; assumption. +cut (Rabs (h / 2) < del). +intro; cut (h / 2 <> 0). +intro; assert (H11 := H2 _ H10 H9). +rewrite Rplus_0_l in H11; rewrite sin_0 in H11. +rewrite Rminus_0_r in H11; apply H11. +unfold Rdiv in |- *; apply prod_neq_R0. +apply H7. +apply Rinv_neq_0_compat; discrR. +apply Rlt_trans with (del / 2). +unfold Rdiv in |- *; rewrite Rabs_mult. +rewrite (Rabs_right (/ 2)). +do 2 rewrite <- (Rmult_comm (/ 2)); apply Rmult_lt_compat_l. +apply Rinv_0_lt_compat; prove_sup0. +apply Rlt_le_trans with (pos delta). +apply H8. +unfold delta in |- *; simpl in |- *; apply Rmin_l. +apply Rle_ge; left; apply Rinv_0_lt_compat; prove_sup0. +rewrite <- (Rplus_0_r (del / 2)); pattern del at 1 in |- *; + rewrite (double_var del); apply Rplus_lt_compat_l; + unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply (cond_pos del). +apply Rinv_0_lt_compat; prove_sup0. +elim H5; intros; assert (H11 := H10 (h / 2)). +rewrite sin_0 in H11; do 2 rewrite Rminus_0_r in H11. +apply H11. +split. +unfold D_x, no_cond in |- *; split. +trivial. +apply (sym_not_eq (A:=R)); unfold Rdiv in |- *; apply prod_neq_R0. +apply H7. +apply Rinv_neq_0_compat; discrR. +apply Rlt_trans with (del_c / 2). +unfold Rdiv in |- *; rewrite Rabs_mult. +rewrite (Rabs_right (/ 2)). +do 2 rewrite <- (Rmult_comm (/ 2)). +apply Rmult_lt_compat_l. +apply Rinv_0_lt_compat; prove_sup0. +apply Rlt_le_trans with (pos delta). +apply H8. +unfold delta in |- *; simpl in |- *; apply Rmin_r. +apply Rle_ge; left; apply Rinv_0_lt_compat; prove_sup0. +rewrite <- (Rplus_0_r (del_c / 2)); pattern del_c at 2 in |- *; + rewrite (double_var del_c); apply Rplus_lt_compat_l. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply H9. +apply Rinv_0_lt_compat; prove_sup0. +rewrite Rmult_minus_distr_l; rewrite Rmult_1_r; unfold Rminus in |- *; + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; + rewrite (Rmult_comm 2); unfold Rdiv, Rsqr in |- *. +repeat rewrite Rmult_assoc. +repeat apply Rmult_eq_compat_l. +rewrite Rinv_mult_distr. +rewrite Rinv_involutive. +apply Rmult_comm. +discrR. +apply H7. +apply Rinv_neq_0_compat; discrR. +pattern h at 2 in |- *; replace h with (2 * (h / 2)). +rewrite (cos_2a_sin (h / 2)). +rewrite cos_0; unfold Rsqr in |- *; ring. +unfold Rdiv in |- *; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m. +discrR. +unfold Rmin in |- *; case (Rle_dec del del_c); intro. +apply (cond_pos del). +elim H5; intros; assumption. +apply continuity_sin. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +Qed. + +(**********) +Theorem derivable_pt_lim_sin : forall x:R, derivable_pt_lim sin x (cos x). +intro; assert (H0 := derivable_pt_lim_sin_0). +assert (H := derivable_pt_lim_cos_0). +unfold derivable_pt_lim in H0, H. +unfold derivable_pt_lim in |- *; intros. +cut (0 < eps / 2); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply H1 | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H0 _ H2); intros alp1 H3. +elim (H _ H2); intros alp2 H4. +set (alp := Rmin alp1 alp2). +cut (0 < alp). +intro; exists (mkposreal _ H5); intros. +replace ((sin (x + h) - sin x) / h - cos x) with + (sin x * ((cos h - 1) / h) + cos x * (sin h / h - 1)). +apply Rle_lt_trans with + (Rabs (sin x * ((cos h - 1) / h)) + Rabs (cos x * (sin h / h - 1))). +apply Rabs_triang. +rewrite (double_var eps); apply Rplus_lt_compat. +apply Rle_lt_trans with (Rabs ((cos h - 1) / h)). +rewrite Rabs_mult; rewrite Rmult_comm; + pattern (Rabs ((cos h - 1) / h)) at 2 in |- *; rewrite <- Rmult_1_r; + apply Rmult_le_compat_l. +apply Rabs_pos. +assert (H8 := SIN_bound x); elim H8; intros. +unfold Rabs in |- *; case (Rcase_abs (sin x)); intro. +rewrite <- (Ropp_involutive 1). +apply Ropp_le_contravar; assumption. +assumption. +cut (Rabs h < alp2). +intro; assert (H9 := H4 _ H6 H8). +rewrite cos_0 in H9; rewrite Rplus_0_l in H9; rewrite Rminus_0_r in H9; + apply H9. +apply Rlt_le_trans with alp. +apply H7. +unfold alp in |- *; apply Rmin_r. +apply Rle_lt_trans with (Rabs (sin h / h - 1)). +rewrite Rabs_mult; rewrite Rmult_comm; + pattern (Rabs (sin h / h - 1)) at 2 in |- *; rewrite <- Rmult_1_r; + apply Rmult_le_compat_l. +apply Rabs_pos. +assert (H8 := COS_bound x); elim H8; intros. +unfold Rabs in |- *; case (Rcase_abs (cos x)); intro. +rewrite <- (Ropp_involutive 1); apply Ropp_le_contravar; assumption. +assumption. +cut (Rabs h < alp1). +intro; assert (H9 := H3 _ H6 H8). +rewrite sin_0 in H9; rewrite Rplus_0_l in H9; rewrite Rminus_0_r in H9; + apply H9. +apply Rlt_le_trans with alp. +apply H7. +unfold alp in |- *; apply Rmin_l. +rewrite sin_plus; unfold Rminus, Rdiv in |- *; + repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l; + repeat rewrite Rmult_assoc; repeat rewrite Rplus_assoc; + apply Rplus_eq_compat_l. +rewrite (Rplus_comm (sin x * (-1 * / h))); repeat rewrite Rplus_assoc; + apply Rplus_eq_compat_l. +rewrite Ropp_mult_distr_r_reverse; rewrite Ropp_mult_distr_l_reverse; + rewrite Rmult_1_r; rewrite Rmult_1_l; rewrite Ropp_mult_distr_r_reverse; + rewrite <- Ropp_mult_distr_l_reverse; apply Rplus_comm. +unfold alp in |- *; unfold Rmin in |- *; case (Rle_dec alp1 alp2); intro. +apply (cond_pos alp1). +apply (cond_pos alp2). +Qed. + +Lemma derivable_pt_lim_cos : forall x:R, derivable_pt_lim cos x (- sin x). +intro; cut (forall h:R, sin (h + PI / 2) = cos h). +intro; replace (- sin x) with (cos (x + PI / 2) * (1 + 0)). +generalize (derivable_pt_lim_comp (id + fct_cte (PI / 2))%F sin); intros. +cut (derivable_pt_lim (id + fct_cte (PI / 2)) x (1 + 0)). +cut (derivable_pt_lim sin ((id + fct_cte (PI / 2))%F x) (cos (x + PI / 2))). +intros; generalize (H0 _ _ _ H2 H1); + replace (comp sin (id + fct_cte (PI / 2))%F) with + (fun x:R => sin (x + PI / 2)); [ idtac | reflexivity ]. +unfold derivable_pt_lim in |- *; intros. +elim (H3 eps H4); intros. +exists x0. +intros; rewrite <- (H (x + h)); rewrite <- (H x); apply H5; assumption. +apply derivable_pt_lim_sin. +apply derivable_pt_lim_plus. +apply derivable_pt_lim_id. +apply derivable_pt_lim_const. +rewrite sin_cos; rewrite <- (Rplus_comm x); ring. +intro; rewrite cos_sin; rewrite Rplus_comm; reflexivity. +Qed. + +Lemma derivable_pt_sin : forall x:R, derivable_pt sin x. +unfold derivable_pt in |- *; intro. +apply existT with (cos x). +apply derivable_pt_lim_sin. +Qed. + +Lemma derivable_pt_cos : forall x:R, derivable_pt cos x. +unfold derivable_pt in |- *; intro. +apply existT with (- sin x). +apply derivable_pt_lim_cos. +Qed. + +Lemma derivable_sin : derivable sin. +unfold derivable in |- *; intro; apply derivable_pt_sin. +Qed. + +Lemma derivable_cos : derivable cos. +unfold derivable in |- *; intro; apply derivable_pt_cos. +Qed. + +Lemma derive_pt_sin : + forall x:R, derive_pt sin x (derivable_pt_sin _) = cos x. +intros; apply derive_pt_eq_0. +apply derivable_pt_lim_sin. +Qed. + +Lemma derive_pt_cos : + forall x:R, derive_pt cos x (derivable_pt_cos _) = - sin x. +intros; apply derive_pt_eq_0. +apply derivable_pt_lim_cos. +Qed.
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