Imported Upstream version 8.0pl1upstream/8.0pl1

1 files changed, 101 insertions, 0 deletions

diff --git a/theories/Sets/Ensembles.v b/theories/Sets/Ensembles.v
new file mode 100755
index 00000000..05afc298
--- /dev/null
+++ b/theories/Sets/Ensembles.v
@@ -0,0 +1,101 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+(****************************************************************************)
+(*                                                                          *)
+(*                         Naive Set Theory in Coq                          *)
+(*                                                                          *)
+(*                     INRIA                        INRIA                   *)
+(*              Rocquencourt                        Sophia-Antipolis        *)
+(*                                                                          *)
+(*                                 Coq V6.1                                 *)
+(*									    *)
+(*			         Gilles Kahn 				    *)
+(*				 Gerard Huet				    *)
+(*									    *)
+(*									    *)
+(*                                                                          *)
+(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks  *)
+(* to the Newton Institute for providing an exceptional work environment    *)
+(* in Summer 1995. Several developments by E. Ledinot were an inspiration.  *)
+(****************************************************************************)
+
+(*i $Id: Ensembles.v,v 1.7.2.1 2004/07/16 19:31:17 herbelin Exp $ i*)
+
+Section Ensembles.
+Variable U : Type.
+
+Definition Ensemble := U -> Prop. 
+
+Definition In (A:Ensemble) (x:U) : Prop := A x.
+
+Definition Included (B C:Ensemble) : Prop := forall x:U, In B x -> In C x.
+
+Inductive Empty_set : Ensemble :=.
+
+Inductive Full_set : Ensemble :=
+    Full_intro : forall x:U, In Full_set x.
+
+(** NB: The following definition builds-in equality of elements in [U] as 
+   Leibniz equality. 
+
+   This may have to be changed if we replace [U] by a Setoid on [U] 
+   with its own equality [eqs], with  
+   [In_singleton: (y: U)(eqs x y) -> (In (Singleton x) y)]. *)
+
+Inductive Singleton (x:U) : Ensemble :=
+    In_singleton : In (Singleton x) x.
+
+Inductive Union (B C:Ensemble) : Ensemble :=
+  | Union_introl : forall x:U, In B x -> In (Union B C) x
+  | Union_intror : forall x:U, In C x -> In (Union B C) x.
+
+Definition Add (B:Ensemble) (x:U) : Ensemble := Union B (Singleton x).
+
+Inductive Intersection (B C:Ensemble) : Ensemble :=
+    Intersection_intro :
+      forall x:U, In B x -> In C x -> In (Intersection B C) x.
+
+Inductive Couple (x y:U) : Ensemble :=
+  | Couple_l : In (Couple x y) x
+  | Couple_r : In (Couple x y) y.
+
+Inductive Triple (x y z:U) : Ensemble :=
+  | Triple_l : In (Triple x y z) x
+  | Triple_m : In (Triple x y z) y
+  | Triple_r : In (Triple x y z) z.
+
+Definition Complement (A:Ensemble) : Ensemble := fun x:U => ~ In A x.
+
+Definition Setminus (B C:Ensemble) : Ensemble :=
+  fun x:U => In B x /\ ~ In C x.
+
+Definition Subtract (B:Ensemble) (x:U) : Ensemble := Setminus B (Singleton x).
+
+Inductive Disjoint (B C:Ensemble) : Prop :=
+    Disjoint_intro : (forall x:U, ~ In (Intersection B C) x) -> Disjoint B C.
+
+Inductive Inhabited (B:Ensemble) : Prop :=
+    Inhabited_intro : forall x:U, In B x -> Inhabited B.
+
+Definition Strict_Included (B C:Ensemble) : Prop := Included B C /\ B <> C.
+
+Definition Same_set (B C:Ensemble) : Prop := Included B C /\ Included C B.
+
+(** Extensionality Axiom *)
+
+Axiom Extensionality_Ensembles : forall A B:Ensemble, Same_set A B -> A = B.
+Hint Resolve Extensionality_Ensembles.
+
+End Ensembles.
+
+Hint Unfold In Included Same_set Strict_Included Add Setminus Subtract: sets
+  v62.
+
+Hint Resolve Union_introl Union_intror Intersection_intro In_singleton
+  Couple_l Couple_r Triple_l Triple_m Triple_r Disjoint_intro
+  Extensionality_Ensembles: sets v62.
\ No newline at end of file