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authorGravatar glondu <glondu@85f007b7-540e-0410-9357-904b9bb8a0f7>2009-09-17 15:58:14 +0000
committerGravatar glondu <glondu@85f007b7-540e-0410-9357-904b9bb8a0f7>2009-09-17 15:58:14 +0000
commit61ccbc81a2f3b4662ed4a2bad9d07d2003dda3a2 (patch)
tree961cc88c714aa91a0276ea9fbf8bc53b2b9d5c28 /theories/Wellfounded/Union.v
parent6d3fbdf36c6a47b49c2a4b16f498972c93c07574 (diff)
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
Diffstat (limited to 'theories/Wellfounded/Union.v')
-rw-r--r--theories/Wellfounded/Union.v10
1 files changed, 5 insertions, 5 deletions
diff --git a/theories/Wellfounded/Union.v b/theories/Wellfounded/Union.v
index ebf4ba98e..fbb3d9e3c 100644
--- a/theories/Wellfounded/Union.v
+++ b/theories/Wellfounded/Union.v
@@ -17,9 +17,9 @@ Require Import Transitive_Closure.
Section WfUnion.
Variable A : Type.
Variables R1 R2 : relation A.
-
+
Notation Union := (union A R1 R2).
-
+
Remark strip_commut :
commut A R1 R2 ->
forall x y:A,
@@ -29,7 +29,7 @@ Section WfUnion.
induction 2 as [x y| x y z H0 IH1 H1 IH2]; intros.
elim H with y x z; auto with sets; intros x0 H2 H3.
exists x0; auto with sets.
-
+
elim IH1 with z0; auto with sets; intros.
elim IH2 with x0; auto with sets; intros.
exists x1; auto with sets.
@@ -50,7 +50,7 @@ Section WfUnion.
elim H8; intros.
apply H6; auto with sets.
apply t_trans with x0; auto with sets.
-
+
elim strip_commut with x x0 y0; auto with sets; intros.
apply Acc_inv_trans with x1; auto with sets.
unfold union in |- *.
@@ -63,7 +63,7 @@ Section WfUnion.
apply Acc_intro; auto with sets.
Qed.
-
+
Theorem wf_union :
commut A R1 R2 -> well_founded R1 -> well_founded R2 -> well_founded Union.
Proof.