Remplacement des fichiers .v ancienne syntaxe de theories, contrib et states par les fichiers nouvelle syntaxe

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@5027 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 45 insertions, 42 deletions

diff --git a/theories/Wellfounded/Union.v b/theories/Wellfounded/Union.v
index ee45a9476..d7f241dd0 100644
--- a/theories/Wellfounded/Union.v
+++ b/theories/Wellfounded/Union.v
@@ -10,65 +10,68 @@
 
 (** Author: Bruno Barras *)
 
-Require Relation_Operators.
-Require Relation_Definitions.
-Require Transitive_Closure.
+Require Import Relation_Operators.
+Require Import Relation_Definitions.
+Require Import Transitive_Closure.
 
 Section WfUnion.
-  Variable A: Set.
-  Variable R1,R2: (relation A).
+  Variable A : Set.
+  Variables R1 R2 : relation A.
   
  Notation Union := (union A R1 R2).
 
- Hints Resolve Acc_clos_trans wf_clos_trans.
+ Hint Resolve Acc_clos_trans wf_clos_trans.
 
-Remark strip_commut: 
-      (commut A R1 R2)->(x,y:A)(clos_trans A R1 y x)->(z:A)(R2 z y)
-            ->(EX y':A | (R2 y' x) & (clos_trans A R1 z y')).
+Remark strip_commut :
+ commut A R1 R2 ->
+ forall x y:A,
+   clos_trans A R1 y x ->
+   forall z:A, R2 z y ->  exists2 y' : A | R2 y' x & clos_trans A R1 z y'.
 Proof.
- NewInduction 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.
+ 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.
- Apply t_trans with 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.
+ apply t_trans with x0; auto with sets.
 Qed.
 
 
-  Lemma Acc_union:  (commut A R1 R2)->((x:A)(Acc A R2 x)->(Acc A R1 x))
-                         ->(a:A)(Acc A R2 a)->(Acc A Union a).
+  Lemma Acc_union :
+   commut A R1 R2 ->
+   (forall x:A, Acc R2 x -> Acc R1 x) -> forall a:A, Acc R2 a -> Acc Union a.
 Proof.
- NewInduction 3 as [x H1 H2].
- Apply Acc_intro;Intros.
- Elim H3;Intros;Auto with sets.
- Cut (clos_trans A  R1 y x);Auto with sets.
- ElimType (Acc A (clos_trans A R1) y);Intros.
- Apply Acc_intro;Intros.
- Elim H8;Intros.
- Apply H6;Auto with sets.
- Apply t_trans with x0 ;Auto with sets.
+ induction 3 as [x H1 H2].
+ apply Acc_intro; intros.
+ elim H3; intros; auto with sets.
+ cut (clos_trans A R1 y x); auto with sets.
+ elimtype (Acc (clos_trans A R1) y); intros.
+ apply Acc_intro; intros.
+ 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 .
- Elim H11;Auto with sets;Intros.
- Apply t_trans with y1 ;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 |- *.
+ elim H11; auto with sets; intros.
+ apply t_trans with y1; auto with sets.
 
- Apply (Acc_clos_trans A).
- Apply Acc_inv with x ;Auto with sets.
- Apply H0.
- Apply Acc_intro;Auto with sets.
+ apply (Acc_clos_trans A).
+ apply Acc_inv with x; auto with sets.
+ apply H0.
+ apply Acc_intro; auto with sets.
 Qed.
 
 
-  Theorem wf_union:  (commut A R1 R2)->(well_founded A R1)->(well_founded A R2)
-                  ->(well_founded A Union).
+  Theorem wf_union :
+   commut A R1 R2 -> well_founded R1 -> well_founded R2 -> well_founded Union.
 Proof.
- Unfold well_founded .
- Intros.
- Apply Acc_union;Auto with sets.
+ unfold well_founded in |- *.
+ intros.
+ apply Acc_union; auto with sets.
 Qed.
 
-End WfUnion.
+End WfUnion.
\ No newline at end of file