Delete trailing whitespaces in all *.{v,ml*} files

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12337 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 25 insertions, 25 deletions

diff --git a/plugins/fourier/Fourier_util.v b/plugins/fourier/Fourier_util.v
index c592af09a..0fd92d606 100644
--- a/plugins/fourier/Fourier_util.v
+++ b/plugins/fourier/Fourier_util.v
@@ -12,17 +12,17 @@ Require Export Rbase.
 Comments "Lemmas used by the tactic Fourier".
 
 Open Scope R_scope.
- 
+
 Lemma Rfourier_lt : forall x1 y1 a:R, x1 < y1 -> 0 < a -> a * x1 < a * y1.
 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 |- *.
 intros.
 case H; auto with real.
 Qed.
- 
+
 Lemma Rfourier_lt_lt :
  forall x1 y1 x2 y2 a:R,
    x1 < y1 -> x2 < y2 -> 0 < a -> x1 + a * x2 < y1 + a * y2.
@@ -33,7 +33,7 @@ apply Rfourier_lt.
 try exact H0.
 try exact H1.
 Qed.
- 
+
 Lemma Rfourier_lt_le :
  forall x1 y1 x2 y2 a:R,
    x1 < y1 -> x2 <= y2 -> 0 < a -> x1 + a * x2 < y1 + a * y2.
@@ -48,7 +48,7 @@ rewrite (Rplus_comm x1 (a * y2)).
 apply Rplus_lt_compat_l.
 try exact H.
 Qed.
- 
+
 Lemma Rfourier_le_lt :
  forall x1 y1 x2 y2 a:R,
    x1 <= y1 -> x2 < y2 -> 0 < a -> x1 + a * x2 < y1 + a * y2.
@@ -59,7 +59,7 @@ rewrite H2.
 apply Rplus_lt_compat_l.
 apply Rfourier_lt; auto with real.
 Qed.
- 
+
 Lemma Rfourier_le_le :
  forall x1 y1 x2 y2 a:R,
    x1 <= y1 -> x2 <= y2 -> 0 < a -> x1 + a * x2 <= y1 + a * y2.
@@ -81,25 +81,25 @@ red in |- *.
 right; try assumption.
 auto with real.
 Qed.
- 
+
 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.
 Qed.
- 
+
 Lemma Rlt_mult_inv_pos : forall x y:R, 0 < x -> 0 < y -> 0 < x * / y.
 intros x y H H0; try assumption.
 replace 0 with (x * 0).
 apply Rmult_lt_compat_l; auto with real.
 ring.
 Qed.
- 
+
 Lemma Rlt_zero_1 : 0 < 1.
 exact Rlt_0_1.
 Qed.
- 
+
 Lemma Rle_zero_pos_plus1 : forall x:R, 0 <= x -> 0 <= 1 + x.
 intros x H; try assumption.
 case H; intros.
@@ -112,7 +112,7 @@ red in |- *; left.
 exact Rlt_zero_1.
 ring.
 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.
@@ -121,12 +121,12 @@ apply Rlt_mult_inv_pos; auto with real.
 rewrite <- H1.
 red in |- *; right; ring.
 Qed.
- 
+
 Lemma Rle_zero_1 : 0 <= 1.
 red in |- *; 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.
 generalize (Rgt_not_le 0 (n * / d)).
@@ -144,14 +144,14 @@ ring.
 ring.
 ring.
 Qed.
- 
+
 Lemma Rnot_lt0 : forall x:R, ~ 0 < 0 * x.
 intros x; try assumption.
 replace (0 * x) with 0.
 apply Rlt_irrefl.
 ring.
 Qed.
- 
+
 Lemma Rlt_not_le_frac_opp : forall n d:R, 0 < n * / d -> ~ 0 <= - n * / d.
 intros n d H; try assumption.
 apply Rgt_not_le.
@@ -162,7 +162,7 @@ try exact H.
 ring.
 ring.
 Qed.
- 
+
 Lemma Rnot_lt_lt : forall x y:R, ~ 0 < y - x -> ~ x < y.
 unfold not in |- *; intros.
 apply H.
@@ -173,7 +173,7 @@ try exact H0.
 ring.
 ring.
 Qed.
- 
+
 Lemma Rnot_le_le : forall x y:R, ~ 0 <= y - x -> ~ x <= y.
 unfold not in |- *; intros.
 apply H.
@@ -188,35 +188,35 @@ ring.
 right.
 rewrite H1; ring.
 Qed.
- 
+
 Lemma Rfourier_gt_to_lt : forall x y:R, y > x -> x < y.
 unfold Rgt in |- *; intros; assumption.
 Qed.
- 
+
 Lemma Rfourier_ge_to_le : forall x y:R, y >= x -> x <= y.
 intros x y; exact (Rge_le y x).
 Qed.
- 
+
 Lemma Rfourier_eqLR_to_le : forall x y:R, x = y -> x <= y.
 exact Req_le.
 Qed.
- 
+
 Lemma Rfourier_eqRL_to_le : forall x y:R, y = x -> x <= y.
 exact Req_le_sym.
 Qed.
- 
+
 Lemma Rfourier_not_ge_lt : forall x y:R, (x >= y -> False) -> x < y.
 exact Rnot_ge_lt.
 Qed.
- 
+
 Lemma Rfourier_not_gt_le : forall x y:R, (x > y -> False) -> x <= y.
 exact Rnot_gt_le.
 Qed.
- 
+
 Lemma Rfourier_not_le_gt : forall x y:R, (x <= y -> False) -> x > y.
 exact Rnot_le_lt.
 Qed.
- 
+
 Lemma Rfourier_not_lt_ge : forall x y:R, (x < y -> False) -> x >= y.
 exact Rnot_lt_ge.
 Qed.