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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 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*)
Section Ensembles.
Variable U: Type.
Definition Ensemble := U -> Prop.
Definition In : Ensemble -> U -> Prop := [A: Ensemble] [x: U] (A x).
Definition Included : Ensemble -> Ensemble -> Prop :=
[B, C: Ensemble] (x: U) (In B x) -> (In C x).
Inductive Empty_set : Ensemble :=
.
Inductive Full_set : Ensemble :=
Full_intro: (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: (x: U) (In B x) -> (In (Union B C) x)
| Union_intror: (x: U) (In C x) -> (In (Union B C) x).
Definition Add : Ensemble -> U -> Ensemble :=
[B: Ensemble] [x: U] (Union B (Singleton x)).
Inductive Intersection [B, C:Ensemble] : Ensemble :=
Intersection_intro:
(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 : Ensemble -> Ensemble :=
[A: Ensemble] [x: U] ~ (In A x).
Definition Setminus : Ensemble -> Ensemble -> Ensemble :=
[B: Ensemble] [C: Ensemble] [x: U] (In B x) /\ ~ (In C x).
Definition Subtract : Ensemble -> U -> Ensemble :=
[B: Ensemble] [x: U] (Setminus B (Singleton x)).
Inductive Disjoint [B, C:Ensemble] : Prop :=
Disjoint_intro: ((x: U) ~ (In (Intersection B C) x)) -> (Disjoint B C).
Inductive Inhabited [B:Ensemble] : Prop :=
Inhabited_intro: (x: U) (In B x) -> (Inhabited B).
Definition Strict_Included : Ensemble -> Ensemble -> Prop :=
[B, C: Ensemble] (Included B C) /\ ~ B == C.
Definition Same_set : Ensemble -> Ensemble -> Prop :=
[B, C: Ensemble] (Included B C) /\ (Included C B).
(** Extensionality Axiom *)
Axiom Extensionality_Ensembles:
(A,B: Ensemble) (Same_set A B) -> A == B.
Hints Resolve Extensionality_Ensembles.
End Ensembles.
Hints Unfold In Included Same_set Strict_Included Add Setminus Subtract : sets v62.
Hints 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.
|
