diff options
Diffstat (limited to 'theories7/Sets/Ensembles.v')
| -rwxr-xr-x | theories7/Sets/Ensembles.v | 108 |
1 files changed, 0 insertions, 108 deletions
diff --git a/theories7/Sets/Ensembles.v b/theories7/Sets/Ensembles.v deleted file mode 100755 index c3a044c0..00000000 --- a/theories7/Sets/Ensembles.v +++ /dev/null @@ -1,108 +0,0 @@ -(************************************************************************) -(* 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. |