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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) *)
(************************************************************************)
Section Relation_Definition.
Variable A : Type.
Definition relation := A -> A -> Prop.
Variable R : relation.
Section General_Properties_of_Relations.
Definition reflexive : Prop := forall x:A, R x x.
Definition transitive : Prop := forall x y z:A, R x y -> R y z -> R x z.
Definition symmetric : Prop := forall x y:A, R x y -> R y x.
Definition antisymmetric : Prop := forall 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 (R1 R2:relation) : Prop :=
forall x y:A, R1 x y -> R2 x y.
Definition same_relation (R1 R2:relation) : Prop :=
inclusion R1 R2 /\ inclusion R2 R1.
Definition commut (R1 R2:relation) : Prop :=
forall x y:A,
R1 y x -> forall z:A, R2 z y -> exists2 y' : A, R2 y' x & R1 z y'.
End Relations_of_Relations.
End Relation_Definition.
Hint Unfold reflexive transitive antisymmetric symmetric: sets.
Hint 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.
Hint Unfold inclusion same_relation commut: sets.
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