(* -*- coding: utf-8 -*- *)
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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
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(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
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(** * Typeclass-based relations, tactics and standard instances  This is the basic theory needed to formalize morphisms and setoids.  Author: Matthieu Sozeau Institution: LRI, CNRS UMR 8623 - University Paris Sud *)
Require Export Corelib.Classes.Init.
Generalizable Variables A B C D R S T U l eqA eqB eqC eqD.
Set Universe Polymorphism.
Definition crelation (A : Type) := A -> A -> Type.
Definition arrow (A B : Type) := A -> B.
Definition flip {A B C : Type} (f : A -> B -> C) := fun x y => f y x.
Definition iffT (A B : Type) := ((A -> B) * (B -> A))%type.
Global Typeclasses Opaque flip arrow iffT.
(** We allow to unfold the [crelation] definition while doing morphism search. *)
Section Defs.
Context {A : Type}.
(** We rebind crelational properties in separate classes to be able to overload each proof. *)
Class Reflexive (R : crelation A) :=
reflexivity : forall x : A, R x x.
Definition complement (R : crelation A) : crelation A :=
fun x y => R x y -> False.
(** Opaque for proof-search. *)
Typeclasses Opaque complement iffT.

(** These are convertible. *)
Lemma complement_inverse R : complement (flip R) = flip (complement R).
Global Instance relation_implication_preorder : PreOrder (@subrelation A).