Library Coq.Classes.Morphisms_Relations



Require Import Relation_Definitions.
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.

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


Require Import List.

Lemma predicate_equivalence_pointwise (l : list Type) :
  Morphism (@predicate_equivalence l ==> pointwise_lifting iff l) id.

Lemma predicate_implication_pointwise (l : list Type) :
  Morphism (@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

Instance relation_equivalence_pointwise :
  Morphism (relation_equivalence ==> pointwise_relation A (pointwise_relation A iff)) id.

Instance subrelation_pointwise :
  Morphism (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)).