From e0d682ec25282a348d35c5b169abafec48555690 Mon Sep 17 00:00:00 2001 From: Stephane Glondu Date: Mon, 20 Aug 2012 18:27:01 +0200 Subject: Imported Upstream version 8.4dfsg --- theories/Reals/Rtrigo_fun.v | 30 +++++++++++++++--------------- 1 file changed, 15 insertions(+), 15 deletions(-) (limited to 'theories/Reals/Rtrigo_fun.v') diff --git a/theories/Reals/Rtrigo_fun.v b/theories/Reals/Rtrigo_fun.v index b7720141..b131b510 100644 --- a/theories/Reals/Rtrigo_fun.v +++ b/theories/Reals/Rtrigo_fun.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* Rabs (/ INR (fact (S n)) * / / INR (fact n))) 0. Proof. - unfold Un_cv in |- *; intros; elim (Rgt_dec eps 1); intro. - split with 0%nat; intros; rewrite (simpl_fact n); unfold R_dist in |- *; + unfold Un_cv; intros; elim (Rgt_dec eps 1); intro. + split with 0%nat; intros; rewrite (simpl_fact n); unfold R_dist; rewrite (Rminus_0_r (Rabs (/ INR (S n)))); rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0). intro; rewrite (Rabs_pos_eq (/ INR (S n))). @@ -39,7 +39,7 @@ Proof. in H4; rewrite (let (H1, H2) := Rmult_ne (/ INR (S n)) in H1) in H4; rewrite (Rmult_comm (/ INR (S n))) in H4; rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H4; - rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (sym_not_equal (O_S n)))) in H4; + rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (not_eq_sym (O_S n)))) in H4; rewrite (let (H1, H2) := Rmult_ne eps in H1) in H4; assumption. apply Rlt_minus; unfold Rgt in a; rewrite <- Rinv_1; @@ -47,11 +47,11 @@ Proof. rewrite (let (H1, H2) := Rmult_ne eps in H2); unfold Rgt in H; assumption. unfold Rgt in H1; apply Rlt_le; assumption. - unfold Rgt in |- *; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. + unfold Rgt; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. (**) cut (0 <= up (/ eps - 1))%Z. intro; elim (IZN (up (/ eps - 1)) H0); intros; split with x; intros; - rewrite (simpl_fact n); unfold R_dist in |- *; + rewrite (simpl_fact n); unfold R_dist; rewrite (Rminus_0_r (Rabs (/ INR (S n)))); rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0). intro; rewrite (Rabs_pos_eq (/ INR (S n))). @@ -72,28 +72,28 @@ Proof. in H6; rewrite (let (H1, H2) := Rmult_ne (/ INR (S n)) in H1) in H6; rewrite (Rmult_comm (/ INR (S n))) in H6; rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H6; - rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (sym_not_equal (O_S n)))) in H6; + rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (not_eq_sym (O_S n)))) in H6; rewrite (let (H1, H2) := Rmult_ne eps in H1) in H6; assumption. - cut (IZR (up (/ eps - 1)) = IZR (Z_of_nat x)); + cut (IZR (up (/ eps - 1)) = IZR (Z.of_nat x)); [ intro | rewrite H1; trivial ]. elim (archimed (/ eps - 1)); intros; clear H6; unfold Rgt in H5; rewrite H4 in H5; rewrite INR_IZR_INZ; assumption. unfold Rgt in H1; apply Rlt_le; assumption. - unfold Rgt in |- *; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. + unfold Rgt; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. apply (le_O_IZR (up (/ eps - 1))); apply (Rle_trans 0 (/ eps - 1) (IZR (up (/ eps - 1)))). - generalize (Rnot_gt_le eps 1 b); clear b; unfold Rle in |- *; intro; elim H0; + generalize (Rnot_gt_le eps 1 b); clear b; unfold Rle; intro; elim H0; clear H0; intro. left; unfold Rgt in H; generalize (Rmult_lt_compat_l (/ eps) eps 1 (Rinv_0_lt_compat eps H) H0); rewrite (Rinv_l eps - (sym_not_eq (Rlt_dichotomy_converse 0 eps (or_introl (0 > eps) H)))) + (not_eq_sym (Rlt_dichotomy_converse 0 eps (or_introl (0 > eps) H)))) ; rewrite (let (H1, H2) := Rmult_ne (/ eps) in H1); - intro; fold (/ eps - 1 > 0) in |- *; apply Rgt_minus; - unfold Rgt in |- *; assumption. - right; rewrite H0; rewrite Rinv_1; apply sym_eq; apply Rminus_diag_eq; auto. + intro; fold (/ eps - 1 > 0); apply Rgt_minus; + unfold Rgt; assumption. + right; rewrite H0; rewrite Rinv_1; symmetry; apply Rminus_diag_eq; auto. elim (archimed (/ eps - 1)); intros; clear H1; unfold Rgt in H0; apply Rlt_le; assumption. Qed. -- cgit v1.2.3