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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** * Morphism instances for relations.
Author: Matthieu Sozeau
Institution: LRI, CNRS UMR 8623 - University Paris Sud
*)
Require Import Relation_Definitions.
Require Import Coq.Classes.Morphisms.
Require Import Coq.Program.Program.
Generalizable Variables A l.
(** Morphisms for relations *)
Instance relation_conjunction_morphism : Proper (relation_equivalence (A:=A) ==>
relation_equivalence ==> relation_equivalence) relation_conjunction.
Proof. firstorder. Qed.
Instance relation_disjunction_morphism : Proper (relation_equivalence (A:=A) ==>
relation_equivalence ==> relation_equivalence) relation_disjunction.
Proof. firstorder. Qed.
(* Predicate equivalence is exactly the same as the pointwise lifting of [iff]. *)
Require Import List.
Lemma predicate_equivalence_pointwise (l : Tlist) :
Proper (@predicate_equivalence l ==> pointwise_lifting iff l) id.
Proof. do 2 red. unfold predicate_equivalence. auto. Qed.
Lemma predicate_implication_pointwise (l : Tlist) :
Proper (@predicate_implication l ==> pointwise_lifting impl l) id.
Proof. do 2 red. unfold predicate_implication. auto. Qed.
(** The instanciation at relation allows to rewrite applications of relations
[R x y] to [R' x y] when [R] and [R'] are in [relation_equivalence]. *)
Instance relation_equivalence_pointwise :
Proper (relation_equivalence ==> pointwise_relation A (pointwise_relation A iff)) id.
Proof. intro. apply (predicate_equivalence_pointwise (Tcons A (Tcons A Tnil))). Qed.
Instance subrelation_pointwise :
Proper (subrelation ==> pointwise_relation A (pointwise_relation A impl)) id.
Proof. intro. apply (predicate_implication_pointwise (Tcons A (Tcons A Tnil))). Qed.
Lemma inverse_pointwise_relation A (R : relation A) :
relation_equivalence (pointwise_relation A (inverse R)) (inverse (pointwise_relation A R)).
Proof. intros. split; firstorder. Qed.
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