diff options
author | Stephane Glondu <steph@glondu.net> | 2013-05-08 18:03:54 +0200 |
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committer | Stephane Glondu <steph@glondu.net> | 2013-05-08 18:03:54 +0200 |
commit | db38bb4ad9aff74576d3b7f00028d48f0447d5bd (patch) | |
tree | 09dafc3e5c7361d3a28e93677eadd2b7237d4f9f /plugins/fourier/Fourier_util.v | |
parent | 6e34b272d789455a9be589e27ad3a998cf25496b (diff) | |
parent | 499a11a45b5711d4eaabe84a80f0ad3ae539d500 (diff) |
Merge branch 'experimental/upstream' into upstream
Diffstat (limited to 'plugins/fourier/Fourier_util.v')
-rw-r--r-- | plugins/fourier/Fourier_util.v | 36 |
1 files changed, 17 insertions, 19 deletions
diff --git a/plugins/fourier/Fourier_util.v b/plugins/fourier/Fourier_util.v index 7c5b5ed7..b10c304c 100644 --- a/plugins/fourier/Fourier_util.v +++ b/plugins/fourier/Fourier_util.v @@ -1,13 +1,11 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(* $Id: Fourier_util.v 14641 2011-11-06 11:59:10Z herbelin $ *) - Require Export Rbase. Comments "Lemmas used by the tactic Fourier". @@ -18,7 +16,7 @@ intros; apply Rmult_lt_compat_l; assumption. Qed. Lemma Rfourier_le : forall x1 y1 a:R, x1 <= y1 -> 0 < a -> a * x1 <= a * y1. -red in |- *. +red. intros. case H; auto with real. Qed. @@ -65,19 +63,19 @@ Lemma Rfourier_le_le : x1 <= y1 -> x2 <= y2 -> 0 < a -> x1 + a * x2 <= y1 + a * y2. intros x1 y1 x2 y2 a H H0 H1; try assumption. case H0; intros. -red in |- *. +red. left; try assumption. apply Rfourier_le_lt; auto with real. rewrite H2. case H; intros. -red in |- *. +red. left; try assumption. rewrite (Rplus_comm x1 (a * y2)). rewrite (Rplus_comm y1 (a * y2)). apply Rplus_lt_compat_l. try exact H3. rewrite H3. -red in |- *. +red. right; try assumption. auto with real. Qed. @@ -86,7 +84,7 @@ Lemma Rlt_zero_pos_plus1 : forall x:R, 0 < x -> 0 < 1 + x. intros x H; try assumption. rewrite Rplus_comm. apply Rle_lt_0_plus_1. -red in |- *; auto with real. +red; auto with real. Qed. Lemma Rlt_mult_inv_pos : forall x y:R, 0 < x -> 0 < y -> 0 < x * / y. @@ -103,12 +101,12 @@ Qed. Lemma Rle_zero_pos_plus1 : forall x:R, 0 <= x -> 0 <= 1 + x. intros x H; try assumption. case H; intros. -red in |- *. +red. left; try assumption. apply Rlt_zero_pos_plus1; auto with real. rewrite <- H0. replace (1 + 0) with 1. -red in |- *; left. +red; left. exact Rlt_zero_1. ring. Qed. @@ -116,28 +114,28 @@ Qed. Lemma Rle_mult_inv_pos : forall x y:R, 0 <= x -> 0 < y -> 0 <= x * / y. intros x y H H0; try assumption. case H; intros. -red in |- *; left. +red; left. apply Rlt_mult_inv_pos; auto with real. rewrite <- H1. -red in |- *; right; ring. +red; right; ring. Qed. Lemma Rle_zero_1 : 0 <= 1. -red in |- *; left. +red; left. exact Rlt_zero_1. Qed. Lemma Rle_not_lt : forall n d:R, 0 <= n * / d -> ~ 0 < - n * / d. -intros n d H; red in |- *; intros H0; try exact H0. +intros n d H; red; intros H0; try exact H0. generalize (Rgt_not_le 0 (n * / d)). intros H1; elim H1; try assumption. replace (n * / d) with (- - (n * / d)). replace 0 with (- -0). replace (- (n * / d)) with (- n * / d). replace (-0) with 0. -red in |- *. +red. apply Ropp_gt_lt_contravar. -red in |- *. +red. exact H0. ring. ring. @@ -164,7 +162,7 @@ ring. Qed. Lemma Rnot_lt_lt : forall x y:R, ~ 0 < y - x -> ~ x < y. -unfold not in |- *; intros. +unfold not; intros. apply H. apply Rplus_lt_reg_r with x. replace (x + 0) with x. @@ -175,7 +173,7 @@ ring. Qed. Lemma Rnot_le_le : forall x y:R, ~ 0 <= y - x -> ~ x <= y. -unfold not in |- *; intros. +unfold not; intros. apply H. case H0; intros. left. @@ -190,7 +188,7 @@ rewrite H1; ring. Qed. Lemma Rfourier_gt_to_lt : forall x y:R, y > x -> x < y. -unfold Rgt in |- *; intros; assumption. +unfold Rgt; intros; assumption. Qed. Lemma Rfourier_ge_to_le : forall x y:R, y >= x -> x <= y. |