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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(*                                                                          *)
(*                         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.  *)
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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.

End Ensembles.

Hint Unfold In Included Same_set Strict_Included Add Setminus Subtract: sets.

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.