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Diffstat (limited to 'theories7/Sets/Ensembles.v')
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diff --git a/theories7/Sets/Ensembles.v b/theories7/Sets/Ensembles.v new file mode 100755 index 00000000..c3a044c0 --- /dev/null +++ b/theories7/Sets/Ensembles.v @@ -0,0 +1,108 @@ +(************************************************************************) +(* 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. |