Library Coq.Classes.Morphisms_Relations

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.

Morphisms for relations

Instance relation_conjunction_morphism : Proper (relation_equivalence (A:=A) ==>
  relation_equivalence ==> relation_equivalence) relation_conjunction.

Instance relation_disjunction_morphism : Proper (relation_equivalence (A:=A) ==>
  relation_equivalence ==> relation_equivalence) relation_disjunction.

Require Import List.

Lemma predicate_equivalence_pointwise (l : Tlist) :
  Proper (@predicate_equivalence l ==> pointwise_lifting iff l) id.

Lemma predicate_implication_pointwise (l : Tlist) :
  Proper (@predicate_implication l ==> pointwise_lifting impl l) id.

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.

Instance subrelation_pointwise :
  Proper (subrelation ==> pointwise_relation A (pointwise_relation A impl)) id.

Lemma inverse_pointwise_relation A (R : relation A) :
  relation_equivalence (pointwise_relation A (inverse R)) (inverse (pointwise_relation A R)).