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(* -*- coq-prog-args: ("-emacs-U" "-top" "Coq.Classes.Morphisms") -*- *)
(************************************************************************)
(* 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 *)
(************************************************************************)
(* Morphism instances for relations.
Author: Matthieu Sozeau
Institution: LRI, CNRS UMR 8623 - UniversitÃcopyright Paris Sud
91405 Orsay, France *)
Require Import Coq.Classes.Morphisms.
Require Import Coq.Program.Program.
(** Morphisms for relations *)
Instance relation_conjunction_morphism : Morphism (relation_equivalence (A:=A) ==>
relation_equivalence ==> relation_equivalence) relation_conjunction.
Proof. firstorder. Qed.
Instance relation_disjunction_morphism : Morphism (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 : list Type) :
Morphism (@predicate_equivalence l ==> pointwise_lifting iff l) id.
Proof. do 2 red. unfold predicate_equivalence. auto. Qed.
Lemma predicate_implication_pointwise (l : list Type) :
Morphism (@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 :
Morphism (relation_equivalence ==> pointwise_relation (A:=A) (pointwise_relation (A:=A) iff)) id.
Proof. intro. apply (predicate_equivalence_pointwise (cons A (cons A nil))). Qed.
Instance subrelation_pointwise :
Morphism (subrelation ==> pointwise_relation (A:=A) (pointwise_relation (A:=A) impl)) id.
Proof. intro. apply (predicate_implication_pointwise (cons A (cons A nil))). Qed.
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