Library Stdlib.Classes.Morphisms_Relations

Morphism instances for relations.

Author: Matthieu Sozeau Institution: LRI, CNRS UMR 8623 - University Paris Sud

Require Import Relation_Definitions.
Require Import Stdlib.Classes.Morphisms.
Require Import Stdlib.Program.Program.

Generalizable Variables A l.

Morphisms for relations

#[global]
Instance relation_conjunction_morphism {A} : Proper (relation_equivalence (A:=A) ==>
  relation_equivalence ==> relation_equivalence) relation_conjunction.

#[global]
Instance relation_disjunction_morphism {A} : Proper (relation_equivalence (A:=A) ==>
  relation_equivalence ==> relation_equivalence) relation_disjunction.

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 instantiation at relation allows rewriting applications of relations R x y to R' x y when R and R' are in relation_equivalence.

#[global]
Instance relation_equivalence_pointwise {A} :
  Proper (relation_equivalence ==> pointwise_relation A (pointwise_relation A iff)) id.

#[global]
Instance subrelation_pointwise {A} :
  Proper (subrelation ==> pointwise_relation A (pointwise_relation A impl)) id.

Lemma flip_pointwise_relation A (R : relation A) :
  relation_equivalence (pointwise_relation A (flip R)) (flip (pointwise_relation A R)).