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authorGravatar Samuel Mimram <samuel.mimram@ens-lyon.org>2004-07-28 21:54:47 +0000
committerGravatar Samuel Mimram <samuel.mimram@ens-lyon.org>2004-07-28 21:54:47 +0000
commit6b649aba925b6f7462da07599fe67ebb12a3460e (patch)
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+(************************************************************************)
+(* 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