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-rw-r--r--theories/Reals/Rtrigo.v134
1 files changed, 67 insertions, 67 deletions
diff --git a/theories/Reals/Rtrigo.v b/theories/Reals/Rtrigo.v
index 0baece39..c637b7ab 100644
--- a/theories/Reals/Rtrigo.v
+++ b/theories/Reals/Rtrigo.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rtrigo.v 9454 2006-12-15 15:30:59Z bgregoir $ i*)
+(*i $Id$ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -19,8 +19,8 @@ Require Export Cos_plus.
Require Import ZArith_base.
Require Import Zcomplements.
Require Import Classical_Prop.
-Open Local Scope nat_scope.
-Open Local Scope R_scope.
+Local Open Scope nat_scope.
+Local Open Scope R_scope.
(** sin_PI2 is the only remaining axiom **)
Axiom sin_PI2 : sin (PI / 2) = 1.
@@ -32,7 +32,7 @@ Proof.
elim (Rlt_irrefl _ H0).
Qed.
-(**********)
+(**********)
Lemma cos_minus : forall x y:R, cos (x - y) = cos x * cos y + sin x * sin y.
Proof.
intros; unfold Rminus in |- *; rewrite cos_plus.
@@ -50,7 +50,7 @@ Lemma cos2 : forall x:R, Rsqr (cos x) = 1 - Rsqr (sin x).
Proof.
intro x; generalize (sin2_cos2 x); intro H1; rewrite <- H1;
unfold Rminus in |- *; rewrite <- (Rplus_comm (Rsqr (cos x)));
- rewrite Rplus_assoc; rewrite Rplus_opp_r; symmetry in |- *;
+ rewrite Rplus_assoc; rewrite Rplus_opp_r; symmetry in |- *;
apply Rplus_0_r.
Qed.
@@ -151,7 +151,7 @@ Proof.
rewrite <- Rinv_r_sym.
rewrite Rmult_1_l; rewrite (Rmult_comm (sin x));
rewrite <- Ropp_mult_distr_r_reverse; repeat rewrite Rmult_assoc;
- apply Rmult_eq_compat_l; rewrite (Rmult_comm (/ cos y));
+ apply Rmult_eq_compat_l; rewrite (Rmult_comm (/ cos y));
rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
apply Rmult_1_r.
assumption.
@@ -185,7 +185,7 @@ Qed.
Lemma cos_2a_cos : forall x:R, cos (2 * x) = 2 * cos x * cos x - 1.
Proof.
intro x; rewrite double; unfold Rminus in |- *; rewrite Rmult_assoc;
- rewrite cos_plus; generalize (sin2_cos2 x); rewrite double;
+ rewrite cos_plus; generalize (sin2_cos2 x); rewrite double;
intro H1; rewrite <- H1; ring_Rsqr.
Qed.
@@ -219,7 +219,7 @@ Qed.
Lemma tan_0 : tan 0 = 0.
Proof.
unfold tan in |- *; rewrite sin_0; rewrite cos_0.
- unfold Rdiv in |- *; apply Rmult_0_l.
+ unfold Rdiv in |- *; apply Rmult_0_l.
Qed.
Lemma tan_neg : forall x:R, tan (- x) = - tan x.
@@ -320,7 +320,7 @@ Lemma PI2_RGT_0 : 0 < PI / 2.
Proof.
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup ].
-Qed.
+Qed.
Lemma SIN_bound : forall x:R, -1 <= sin x <= 1.
Proof.
@@ -331,13 +331,13 @@ Proof.
intro;
generalize
(Rsqr_incrst_1 1 (sin x) H (Rlt_le 0 1 Rlt_0_1)
- (Rlt_le 0 (sin x) (Rlt_trans 0 1 (sin x) Rlt_0_1 H)));
+ (Rlt_le 0 (sin x) (Rlt_trans 0 1 (sin x) Rlt_0_1 H)));
rewrite Rsqr_1; intro; rewrite sin2 in H0; unfold Rminus in H0;
generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0);
- repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l;
+ repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l;
rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1;
generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1);
- repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x));
+ repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x));
intro; elim (Rlt_irrefl 0 (Rle_lt_trans 0 (Rsqr (cos x)) 0 H3 H2)).
auto with real.
cut (sin x < -1).
@@ -346,13 +346,13 @@ Proof.
generalize
(Rsqr_incrst_1 1 (- sin x) H (Rlt_le 0 1 Rlt_0_1)
(Rlt_le 0 (- sin x) (Rlt_trans 0 1 (- sin x) Rlt_0_1 H)));
- rewrite Rsqr_1; intro; rewrite <- Rsqr_neg in H0;
+ rewrite Rsqr_1; intro; rewrite <- Rsqr_neg in H0;
rewrite sin2 in H0; unfold Rminus in H0;
generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0);
- repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l;
+ repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l;
rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1;
generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1);
- repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x));
+ repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x));
intro; elim (Rlt_irrefl 0 (Rle_lt_trans 0 (Rsqr (cos x)) 0 H3 H2)).
auto with real.
Qed.
@@ -366,7 +366,7 @@ Lemma cos_sin_0 : forall x:R, ~ (cos x = 0 /\ sin x = 0).
Proof.
intro; red in |- *; intro; elim H; intros; generalize (sin2_cos2 x); intro;
rewrite H0 in H2; rewrite H1 in H2; repeat rewrite Rsqr_0 in H2;
- rewrite Rplus_0_r in H2; generalize Rlt_0_1; intro;
+ rewrite Rplus_0_r in H2; generalize Rlt_0_1; intro;
rewrite <- H2 in H3; elim (Rlt_irrefl 0 H3).
Qed.
@@ -399,18 +399,18 @@ Proof.
repeat rewrite Rmult_1_l; repeat rewrite Rmult_1_r;
replace (-1 * Un 1%nat) with (- Un 1%nat); [ idtac | ring ];
replace (-1 * -1 * Un 2%nat) with (Un 2%nat); [ idtac | ring ];
- replace (-1 * (-1 * -1) * Un 3%nat) with (- Un 3%nat);
+ replace (-1 * (-1 * -1) * Un 3%nat) with (- Un 3%nat);
[ idtac | ring ];
replace (Un 0%nat + - Un 1%nat + Un 2%nat + - Un 3%nat) with
(Un 0%nat - Un 1%nat + (Un 2%nat - Un 3%nat)); [ idtac | ring ].
apply Rplus_lt_0_compat.
unfold Rminus in |- *; apply Rplus_lt_reg_r with (Un 1%nat);
- rewrite Rplus_0_r; rewrite (Rplus_comm (Un 1%nat));
- rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
+ rewrite Rplus_0_r; rewrite (Rplus_comm (Un 1%nat));
+ rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
apply H1.
unfold Rminus in |- *; apply Rplus_lt_reg_r with (Un 3%nat);
- rewrite Rplus_0_r; rewrite (Rplus_comm (Un 3%nat));
- rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
+ rewrite Rplus_0_r; rewrite (Rplus_comm (Un 3%nat));
+ rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
apply H1.
intro; unfold Un in |- *.
cut ((2 * S n + 1)%nat = (2 * n + 1 + 2)%nat).
@@ -533,7 +533,7 @@ Proof.
(SIN (PI - x) (Rlt_le 0 (PI - x) H7)
(Rlt_le (PI - x) PI (Rlt_trans (PI - x) (PI / 2) PI H5 PI2_Rlt_PI)));
intros H8 _;
- generalize (sin_lb_gt_0 (PI - x) H7 (Rlt_le (PI - x) (PI / 2) H5));
+ generalize (sin_lb_gt_0 (PI - x) H7 (Rlt_le (PI - x) (PI / 2) H5));
intro H9; apply (Rlt_le_trans 0 (sin_lb (PI - x)) (sin (PI - x)) H9 H8).
reflexivity.
pattern PI at 2 in |- *; rewrite double_var; ring.
@@ -545,7 +545,7 @@ Proof.
intros; rewrite cos_sin;
generalize (Rplus_lt_compat_l (PI / 2) (- (PI / 2)) x H).
rewrite Rplus_opp_r; intro H1;
- generalize (Rplus_lt_compat_l (PI / 2) x (PI / 2) H0);
+ generalize (Rplus_lt_compat_l (PI / 2) x (PI / 2) H0);
rewrite <- double_var; intro H2; apply (sin_gt_0 (PI / 2 + x) H1 H2).
Qed.
@@ -599,7 +599,7 @@ Proof.
replace (PI / 2) with (- PI + 3 * (PI / 2)).
apply Rplus_le_compat_l; assumption.
pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr;
- ring.
+ ring.
unfold INR in |- *; ring.
Qed.
@@ -625,7 +625,7 @@ Proof.
intros; generalize (Rplus_lt_compat_l (2 * PI) (- PI) x H);
replace (2 * PI + - PI) with PI;
[ intro H1; rewrite Rplus_comm in H1;
- generalize (Rplus_lt_compat_l (2 * PI) x 0 H0);
+ generalize (Rplus_lt_compat_l (2 * PI) x 0 H0);
intro H2; rewrite (Rplus_comm (2 * PI)) in H2;
rewrite <- (Rplus_comm 0) in H2; rewrite Rplus_0_l in H2;
rewrite <- (sin_period x 1); unfold INR in |- *;
@@ -644,12 +644,12 @@ Proof.
unfold Rminus in |- *; rewrite (Rplus_comm x); apply Rplus_lt_compat_l;
assumption.
pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr;
- ring.
+ ring.
unfold Rminus in |- *; rewrite Rplus_comm;
replace (PI / 2) with (- PI + 3 * (PI / 2)).
apply Rplus_lt_compat_l; assumption.
pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr;
- ring.
+ ring.
unfold INR in |- *; ring.
Qed.
@@ -658,7 +658,7 @@ Proof.
intros x H1 H2; unfold tan in |- *; generalize _PI2_RLT_0;
generalize (Rlt_trans 0 x (PI / 2) H1 H2); intros;
generalize (Rlt_trans (- (PI / 2)) 0 x H0 H1); intro H5;
- generalize (Rlt_trans x (PI / 2) PI H2 PI2_Rlt_PI);
+ generalize (Rlt_trans x (PI / 2) PI H2 PI2_Rlt_PI);
intro H7; unfold Rdiv in |- *; apply Rmult_lt_0_compat.
apply sin_gt_0; assumption.
apply Rinv_0_lt_compat; apply cos_gt_0; assumption.
@@ -667,7 +667,7 @@ Qed.
Lemma tan_lt_0 : forall x:R, - (PI / 2) < x -> x < 0 -> tan x < 0.
Proof.
intros x H1 H2; unfold tan in |- *;
- generalize (cos_gt_0 x H1 (Rlt_trans x 0 (PI / 2) H2 PI2_RGT_0));
+ generalize (cos_gt_0 x H1 (Rlt_trans x 0 (PI / 2) H2 PI2_RGT_0));
intro H3; rewrite <- Ropp_0;
replace (sin x / cos x) with (- (- sin x / cos x)).
rewrite <- sin_neg; apply Ropp_gt_lt_contravar;
@@ -688,11 +688,11 @@ Proof.
intros; rewrite <- cos_neg; rewrite <- (cos_period (- x) 1);
unfold INR in |- *; replace (- x + 2 * 1 * PI) with (2 * PI - x).
generalize (Ropp_le_ge_contravar x (2 * PI) H0); intro H1;
- generalize (Rge_le (- x) (- (2 * PI)) H1); clear H1;
+ generalize (Rge_le (- x) (- (2 * PI)) H1); clear H1;
intro H1; generalize (Rplus_le_compat_l (2 * PI) (- (2 * PI)) (- x) H1).
- rewrite Rplus_opp_r.
+ rewrite Rplus_opp_r.
intro H2; generalize (Ropp_le_ge_contravar (3 * (PI / 2)) x H); intro H3;
- generalize (Rge_le (- (3 * (PI / 2))) (- x) H3); clear H3;
+ generalize (Rge_le (- (3 * (PI / 2))) (- x) H3); clear H3;
intro H3;
generalize (Rplus_le_compat_l (2 * PI) (- x) (- (3 * (PI / 2))) H3).
replace (2 * PI + - (3 * (PI / 2))) with (PI / 2).
@@ -780,11 +780,11 @@ Proof.
generalize
(Rmult_le_compat_l (/ 2) (x - y) PI
(Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H8).
- repeat rewrite (Rmult_comm (/ 2)).
+ repeat rewrite (Rmult_comm (/ 2)).
intro H9;
generalize
(sin_gt_0 ((x - y) / 2) H6
- (Rle_lt_trans ((x - y) / 2) (PI / 2) PI H9 PI2_Rlt_PI));
+ (Rle_lt_trans ((x - y) / 2) (PI / 2) PI H9 PI2_Rlt_PI));
intro H10;
elim
(Rlt_irrefl (sin ((x - y) / 2))
@@ -799,7 +799,7 @@ Proof.
generalize
(Rmult_le_compat_l (/ 2) (x + y) PI
(Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H4).
- repeat rewrite (Rmult_comm (/ 2)).
+ repeat rewrite (Rmult_comm (/ 2)).
clear H4; intro H4;
generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) y H H1);
replace (- (PI / 2) + - (PI / 2)) with (- PI).
@@ -813,7 +813,7 @@ Proof.
elim H5; intro H50.
generalize (cos_gt_0 ((x + y) / 2) H50 H40); intro H6;
generalize (Rmult_lt_compat_l 2 0 (cos ((x + y) / 2)) Hyp H6).
- rewrite Rmult_0_r.
+ rewrite Rmult_0_r.
clear H6; intro H6; case (Rcase_abs (sin ((x - y) / 2))); intro H7.
assumption.
generalize (Rge_le (sin ((x - y) / 2)) 0 H7); clear H7; intro H7;
@@ -824,7 +824,7 @@ Proof.
(Rle_lt_trans 0 (2 * cos ((x + y) / 2) * sin ((x - y) / 2)) 0 H8 H3);
intro H9; elim (Rlt_irrefl 0 H9).
rewrite <- H50 in H3; rewrite cos_neg in H3; rewrite cos_PI2 in H3;
- rewrite Rmult_0_r in H3; rewrite Rmult_0_l in H3;
+ rewrite Rmult_0_r in H3; rewrite Rmult_0_l in H3;
elim (Rlt_irrefl 0 H3).
unfold Rdiv in H3.
rewrite H40 in H3; assert (H50 := cos_PI2); unfold Rdiv in H50;
@@ -865,8 +865,8 @@ Proof.
clear H5 H6 H7; intro H5; generalize (Ropp_le_ge_contravar (- (PI / 2)) y H1);
rewrite Ropp_involutive; clear H1; intro H1;
generalize (Rge_le (PI / 2) (- y) H1); clear H1; intro H1;
- generalize (Ropp_le_ge_contravar y (PI / 2) H2); clear H2;
- intro H2; generalize (Rge_le (- y) (- (PI / 2)) H2);
+ generalize (Ropp_le_ge_contravar y (PI / 2) H2); clear H2;
+ intro H2; generalize (Rge_le (- y) (- (PI / 2)) H2);
clear H2; intro H2; generalize (Rplus_lt_compat_l (- y) x y H3);
replace (- y + x) with (x - y).
rewrite Rplus_opp_l.
@@ -885,12 +885,12 @@ Proof.
replace (/ 2 * (x - y)) with ((x - y) / 2).
clear H7; intro H7; clear H H0 H1 H2; apply Rminus_lt; rewrite form4;
generalize (cos_gt_0 ((x + y) / 2) H4 H5); intro H8;
- generalize (Rmult_lt_0_compat 2 (cos ((x + y) / 2)) Hyp H8);
+ generalize (Rmult_lt_0_compat 2 (cos ((x + y) / 2)) Hyp H8);
clear H8; intro H8; cut (- PI < - (PI / 2)).
intro H9;
generalize
(sin_lt_0_var ((x - y) / 2)
- (Rlt_le_trans (- PI) (- (PI / 2)) ((x - y) / 2) H9 H7) H6);
+ (Rlt_le_trans (- PI) (- (PI / 2)) ((x - y) / 2) H9 H7) H6);
intro H10;
generalize
(Rmult_lt_gt_compat_neg_l (sin ((x - y) / 2)) 0 (
@@ -1012,21 +1012,21 @@ Proof.
replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)).
apply (sin_increasing_0 (x - 3 * (PI / 2)) (y - 3 * (PI / 2)) H4 H3 H2 H1 H5).
unfold Rminus in |- *.
- rewrite Ropp_mult_distr_l_reverse.
- apply Rplus_comm.
+ rewrite Ropp_mult_distr_l_reverse.
+ apply Rplus_comm.
unfold Rminus in |- *.
- rewrite Ropp_mult_distr_l_reverse.
- apply Rplus_comm.
+ rewrite Ropp_mult_distr_l_reverse.
+ apply Rplus_comm.
pattern PI at 3 in |- *; rewrite double_var.
ring.
rewrite double; pattern PI at 3 4 in |- *; rewrite double_var.
ring.
unfold Rminus in |- *.
- rewrite Ropp_mult_distr_l_reverse.
- apply Rplus_comm.
+ rewrite Ropp_mult_distr_l_reverse.
+ apply Rplus_comm.
unfold Rminus in |- *.
- rewrite Ropp_mult_distr_l_reverse.
- apply Rplus_comm.
+ rewrite Ropp_mult_distr_l_reverse.
+ apply Rplus_comm.
rewrite Rmult_1_r.
rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var.
ring.
@@ -1110,7 +1110,7 @@ Lemma tan_diff :
cos x <> 0 -> cos y <> 0 -> tan x - tan y = sin (x - y) / (cos x * cos y).
Proof.
intros; unfold tan in |- *; rewrite sin_minus.
- unfold Rdiv in |- *.
+ unfold Rdiv in |- *.
unfold Rminus in |- *.
rewrite Rmult_plus_distr_r.
rewrite Rinv_mult_distr.
@@ -1143,7 +1143,7 @@ Lemma tan_increasing_0 :
x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> tan x < tan y -> x < y.
Proof.
intros; generalize PI4_RLT_PI2; intro H4;
- generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4);
+ generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4);
intro H5; change (- (PI / 2) < - (PI / 4)) in H5;
generalize
(cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
@@ -1155,20 +1155,20 @@ Proof.
(sym_not_eq
(Rlt_not_eq 0 (cos x)
(cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
- (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4))));
+ (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4))));
intro H6;
generalize
(sym_not_eq
(Rlt_not_eq 0 (cos y)
(cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
- (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4))));
+ (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4))));
intro H7; generalize (tan_diff x y H6 H7); intro H8;
- generalize (Rlt_minus (tan x) (tan y) H3); clear H3;
+ generalize (Rlt_minus (tan x) (tan y) H3); clear H3;
intro H3; rewrite H8 in H3; cut (sin (x - y) < 0).
intro H9; generalize (Ropp_le_ge_contravar (- (PI / 4)) y H1);
rewrite Ropp_involutive; intro H10; generalize (Rge_le (PI / 4) (- y) H10);
clear H10; intro H10; generalize (Ropp_le_ge_contravar y (PI / 4) H2);
- intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11);
+ intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11);
clear H11; intro H11;
generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11);
generalize (Rplus_le_compat x (PI / 4) (- y) (PI / 4) H0 H10);
@@ -1180,7 +1180,7 @@ Proof.
(sin_gt_0 (x - y) H14 (Rle_lt_trans (x - y) (PI / 2) PI H12 PI2_Rlt_PI));
intro H15; elim (Rlt_irrefl 0 (Rlt_trans 0 (sin (x - y)) 0 H15 H9)).
elim H14; intro H15.
- rewrite <- H15 in H9; rewrite sin_0 in H9; elim (Rlt_irrefl 0 H9).
+ rewrite <- H15 in H9; rewrite sin_0 in H9; elim (Rlt_irrefl 0 H9).
apply Rminus_lt; assumption.
pattern PI at 1 in |- *; rewrite double_var.
unfold Rdiv in |- *.
@@ -1218,7 +1218,7 @@ Proof.
elim
(Rlt_irrefl 0 (Rle_lt_trans 0 (sin (x - y) * / (cos x * cos y)) 0 H13 H3)).
rewrite Rinv_mult_distr.
- reflexivity.
+ reflexivity.
assumption.
assumption.
Qed.
@@ -1229,7 +1229,7 @@ Lemma tan_increasing_1 :
x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> x < y -> tan x < tan y.
Proof.
intros; apply Rminus_lt; generalize PI4_RLT_PI2; intro H4;
- generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4);
+ generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4);
intro H5; change (- (PI / 2) < - (PI / 4)) in H5;
generalize
(cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
@@ -1241,27 +1241,27 @@ Proof.
(sym_not_eq
(Rlt_not_eq 0 (cos x)
(cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
- (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4))));
+ (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4))));
intro H6;
generalize
(sym_not_eq
(Rlt_not_eq 0 (cos y)
(cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
- (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4))));
+ (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4))));
intro H7; rewrite (tan_diff x y H6 H7);
generalize (Rinv_0_lt_compat (cos x) HP1); intro H10;
generalize (Rinv_0_lt_compat (cos y) HP2); intro H11;
generalize (Rmult_lt_0_compat (/ cos x) (/ cos y) H10 H11);
replace (/ cos x * / cos y) with (/ (cos x * cos y)).
clear H10 H11; intro H8; generalize (Ropp_le_ge_contravar y (PI / 4) H2);
- intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11);
+ intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11);
clear H11; intro H11;
generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11);
replace (x + - y) with (x - y).
replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)).
clear H11; intro H9; generalize (Rlt_minus x y H3); clear H3; intro H3;
- clear H H0 H1 H2 H4 H5 HP1 HP2; generalize PI2_Rlt_PI;
- intro H1; generalize (Ropp_lt_gt_contravar (PI / 2) PI H1);
+ clear H H0 H1 H2 H4 H5 HP1 HP2; generalize PI2_Rlt_PI;
+ intro H1; generalize (Ropp_lt_gt_contravar (PI / 2) PI H1);
clear H1; intro H1;
generalize
(sin_lt_0_var (x - y) (Rlt_le_trans (- PI) (- (PI / 2)) (x - y) H1 H9) H3);
@@ -1576,13 +1576,13 @@ Proof.
Qed.
Lemma cos_eq_0_0 :
- forall x:R, cos x = 0 -> exists k : Z, x = IZR k * PI + PI / 2.
+ forall x:R, cos x = 0 -> exists k : Z, x = IZR k * PI + PI / 2.
Proof.
intros x H; rewrite cos_sin in H; generalize (sin_eq_0_0 (PI / INR 2 + x) H);
intro H2; elim H2; intros x0 H3; exists (x0 - Z_of_nat 1)%Z;
rewrite <- Z_R_minus; simpl.
unfold INR in H3. field_simplify [(sym_eq H3)]. field.
-(**
+(**
ring_simplify.
(* rewrite (Rmult_comm PI);*) (* old ring compat *)
rewrite <- H3; simpl;
@@ -1618,7 +1618,7 @@ Proof.
(Rlt_le 0 (/ PI) (Rinv_0_lt_compat PI PI_RGT_0)) H0);
repeat rewrite Rmult_assoc; repeat rewrite <- Rinv_r_sym.
repeat rewrite Rmult_1_r; intro;
- generalize (Rplus_lt_compat_l (IZR (-2)) 1 (IZR k0) H5);
+ generalize (Rplus_lt_compat_l (IZR (-2)) 1 (IZR k0) H5);
rewrite <- plus_IZR.
replace (IZR (-2) + 1) with (-1).
intro; generalize (Rplus_le_compat_l (IZR (-2)) (IZR k0) 2 H6);
@@ -1710,7 +1710,7 @@ Proof.
apply Rplus_le_le_0_compat.
left; unfold Rdiv in |- *; apply Rmult_lt_0_compat.
apply PI_RGT_0.
- apply Rinv_0_lt_compat; prove_sup0.
+ apply Rinv_0_lt_compat; prove_sup0.
assumption.
elim H2; intro.
right; assumption.
generated by cgit v1.2.3 (git 2.39.1) at 2025-01-17 08:23:03 +0000