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authorGravatar Stephane Glondu <steph@glondu.net>2012-08-20 18:27:01 +0200
committerGravatar Stephane Glondu <steph@glondu.net>2012-08-20 18:27:01 +0200
commite0d682ec25282a348d35c5b169abafec48555690 (patch)
tree1a46f0142a85df553388c932110793881f3af52f /plugins/fourier/Fourier_util.v
parent86535d84cc3cffeee1dcd8545343f234e7285530 (diff)
Imported Upstream version 8.4dfsgupstream/8.4dfsg
Diffstat (limited to 'plugins/fourier/Fourier_util.v')
-rw-r--r--plugins/fourier/Fourier_util.v34
1 files changed, 17 insertions, 17 deletions
diff --git a/plugins/fourier/Fourier_util.v b/plugins/fourier/Fourier_util.v
index 3d16f189..b10c304c 100644
--- a/plugins/fourier/Fourier_util.v
+++ b/plugins/fourier/Fourier_util.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
+(* <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 *)
@@ -16,7 +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.
@@ -63,19 +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.
@@ -84,7 +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.
@@ -101,12 +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.
@@ -114,28 +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.
@@ -162,7 +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.
@@ -173,7 +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.
@@ -188,7 +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.