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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 *)
(************************************************************************)
(*i $Id: Relation_Definitions.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*)
Section Relation_Definition.
Variable A: Type.
Definition relation := A -> A -> Prop.
Variable R: relation.
Section General_Properties_of_Relations.
Definition reflexive : Prop := (x: A) (R x x).
Definition transitive : Prop := (x,y,z: A) (R x y) -> (R y z) -> (R x z).
Definition symmetric : Prop := (x,y: A) (R x y) -> (R y x).
Definition antisymmetric : Prop := (x,y: A) (R x y) -> (R y x) -> x=y.
(* for compatibility with Equivalence in ../PROGRAMS/ALG/ *)
Definition equiv := reflexive /\ transitive /\ symmetric.
End General_Properties_of_Relations.
Section Sets_of_Relations.
Record preorder : Prop := {
preord_refl : reflexive;
preord_trans : transitive }.
Record order : Prop := {
ord_refl : reflexive;
ord_trans : transitive;
ord_antisym : antisymmetric }.
Record equivalence : Prop := {
equiv_refl : reflexive;
equiv_trans : transitive;
equiv_sym : symmetric }.
Record PER : Prop := {
per_sym : symmetric;
per_trans : transitive }.
End Sets_of_Relations.
Section Relations_of_Relations.
Definition inclusion : relation -> relation -> Prop :=
[R1,R2: relation] (x,y:A) (R1 x y) -> (R2 x y).
Definition same_relation : relation -> relation -> Prop :=
[R1,R2: relation] (inclusion R1 R2) /\ (inclusion R2 R1).
Definition commut : relation -> relation -> Prop :=
[R1,R2:relation] (x,y:A) (R1 y x) -> (z:A) (R2 z y)
-> (EX y':A |(R2 y' x) & (R1 z y')).
End Relations_of_Relations.
End Relation_Definition.
Hints Unfold reflexive transitive antisymmetric symmetric : sets v62.
Hints Resolve Build_preorder Build_order Build_equivalence
Build_PER preord_refl preord_trans
ord_refl ord_trans ord_antisym
equiv_refl equiv_trans equiv_sym
per_sym per_trans : sets v62.
Hints Unfold inclusion same_relation commut : sets v62.
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