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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) *)
(************************************************************************)
(****************************************************************************)
(* *)
(* 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. *)
(****************************************************************************)
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.
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