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.

Definition RelCompFun {A} {B : Type}(R:relation B)(f:A->B) : relation A :=
 fun a a' => R (f a) (f a').

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.

Class Measure {A B} (f : A -> B).

#[global]
Instance fst_measure {A B} : @Measure (A * B) A Fst := {}.

#[global]
Instance snd_measure {A B} : @Measure (A * B) B Snd := {}.

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.