The Coq Proof Assistant

# Library Coq.Classes.RelationPairs

# Relations over pairs

Require Import SetoidList.
Require Import Relations Morphisms.
Local Notation Fst := (@fst _ _).
Local Notation Snd := (@snd _ _).
Arguments relation_conjunction A%_type (R R')%_signature _ _.
Arguments relation_equivalence A%_type (_ _)%_signature.
Arguments subrelation A%_type (R R')%_signature.
Arguments Reflexive A%_type R%_signature.
Arguments Irreflexive A%_type R%_signature.
Arguments Symmetric A%_type R%_signature.
Arguments Transitive A%_type R%_signature.
Arguments PER A%_type R%_signature.
Arguments Equivalence A%_type R%_signature.
Arguments StrictOrder A%_type R%_signature.
Generalizable Variables A B RA RB Ri Ro f.
Any function from A to B allow to obtain a relation over A out of a relation over B.
Definition RelCompFun {A} {B : Type}(R:relation B)(f:A->B) : relation A :=
fun a a' => R (f a) (f a').
Instances on RelCompFun must match syntactically
Global Typeclasses Opaque RelCompFun.
Infix "@@" := RelCompFun (at level 30, right associativity) : signature_scope.
Notation "R @@1" := (R @@ Fst)%signature (at level 30) : signature_scope.
Notation "R @@2" := (R @@ Snd)%signature (at level 30) : signature_scope.
We declare measures to the system using the Measure class. Otherwise the instances would easily introduce loops, never instantiating the f function.
Class Measure {A B} (f : A -> B).
Standard measures.
#[global]
Instance fst_measure {A B} : @Measure (A * B) A Fst := {}.
#[global]
Instance snd_measure {A B} : @Measure (A * B) B Snd := {}.
We define a product relation over A*B: each components should satisfy the corresponding initial relation.
Definition RelProd {A : Type} {B : Type} (RA:relation A)(RB:relation B) : relation (A*B) :=
relation_conjunction (@RelCompFun (A * B) A RA fst) (RB @@2).
Global Typeclasses Opaque RelProd.
Infix "*" := RelProd : signature_scope.
Section RelCompFun_Instances.
Context {A : Type} {B : Type} (R : relation B).
Global Instance RelCompFun_Reflexive
`(Measure A B f, Reflexive _ R) : Reflexive (R@@f).
Global Instance RelCompFun_Symmetric
`(Measure A B f, Symmetric _ R) : Symmetric (R@@f).
Global Instance RelCompFun_Transitive
`(Measure A B f, Transitive _ R) : Transitive (R@@f).
Global Instance RelCompFun_Irreflexive
`(Measure A B f, Irreflexive _ R) : Irreflexive (R@@f).
Global Instance RelCompFun_Equivalence
`(Measure A B f, Equivalence _ R) : Equivalence (R@@f) := {}.
Global Instance RelCompFun_StrictOrder
`(Measure A B f, StrictOrder _ R) : StrictOrder (R@@f) := {}.
End RelCompFun_Instances.
Section RelProd_Instances.
Context {A : Type} {B : Type} (RA : relation A) (RB : relation B).
Global Instance RelProd_Reflexive `(Reflexive _ RA, Reflexive _ RB) : Reflexive (RA*RB).
Global Instance RelProd_Symmetric `(Symmetric _ RA, Symmetric _ RB)
: Symmetric (RA*RB).
Global Instance RelProd_Transitive
`(Transitive _ RA, Transitive _ RB) : Transitive (RA*RB).
Global Program Instance RelProd_Equivalence
`(Equivalence _ RA, Equivalence _ RB) : Equivalence (RA*RB).
Lemma FstRel_ProdRel :
relation_equivalence (RA @@1) (RA*(fun _ _ : B => True)).
Lemma SndRel_ProdRel :
relation_equivalence (RB @@2) ((fun _ _ : A =>True) * RB).
Global Instance FstRel_sub :
subrelation (RA*RB) (RA @@1).
Global Instance SndRel_sub :
subrelation (RA*RB) (RB @@2).
Global Instance pair_compat :
Proper (RA==>RB==> RA*RB) (@pair _ _).
Global Instance fst_compat :
Proper (RA*RB ==> RA) Fst.
Global Instance snd_compat :
Proper (RA*RB ==> RB) Snd.
Global Instance RelCompFun_compat (f:A->B)
`(Proper _ (Ri==>Ri==>Ro) RB) :
Proper (Ri@@f==>Ri@@f==>Ro) (RB@@f)%signature.
End RelProd_Instances.
#[global]
Hint Unfold RelProd RelCompFun : core.
#[global]
Hint Extern 2 (RelProd _ _ _ _) => split : core.
#[export] Instance Proper_RelProd_flip_impl: forall A B RA1 RA2 RB1 RB2 (RA : relation A) (RB : relation B),
Proper (RA1 ==> RA2 ==> Basics.flip Basics.impl) RA
-> Proper (RB1 ==> RB2 ==> Basics.flip Basics.impl) RB
-> Proper (RA1 * RB1 ==> RA2 * RB2 ==> Basics.flip Basics.impl) (RA * RB)%signature.
#[export] Instance Proper_RelProd_impl: forall A B RA1 RA2 RB1 RB2 (RA : relation A) (RB : relation B),
Proper (RA1 ==> RA2 ==> Basics.impl) RA
-> Proper (RB1 ==> RB2 ==> Basics.impl) RB
-> Proper (RA1 * RB1 ==> RA2 * RB2 ==> Basics.impl) (RA * RB)%signature.
#[export] Instance Proper_RelProd_iff: forall A B RA1 RA2 RB1 RB2 (RA : relation A) (RB : relation B),
Proper (RA1 ==> RA2 ==> iff) RA
-> Proper (RB1 ==> RB2 ==> iff) RB
-> Proper (RA1 * RB1 ==> RA2 * RB2 ==> iff) (RA * RB)%signature.