SetoidRewrite.v

(* -*- mode: coq; mode: visual-line -*-  *)

(* Typeclass instances to allow rewriting in categories. Examples are given later in the file. *)

(* Init.Tactics contains the definition of the Coq stdlib typeclass_inferences database. It must be imported before Basics.Overture. *)

(** Warning: This imports Coq.Setoids.Setoid from the standard library. Currently the setoid rewriting machinery requires this to work, it depends on this file explicitly. This imports the whole standard library into the namespace.

All files that import WildCat/SetoidRewrite.v will also recursively import the entire Coq.Init standard library.  *)

(** Because of this, this file needs to be the *first* file Require'd in any file that uses it.  Otherwise, the typeclasses hintdb is cleared, breaking typeclass inference.  Moreover, if Foo Requires this file, then Foo must also be the first file Require'd in any file that Requires Foo, and so on. In the long term it would be good if this could be avoided.*)

From Coq Require Init.Tactics.

From HoTT Require Import Basics.Overture Basics.Tactics.

From HoTT Require Import Types.Forall.

From Coq Require Setoids.Setoid.

Import CMorphisms.ProperNotations.

From HoTT Require Import WildCat.Core
  WildCat.NatTrans WildCat.Equiv.


#[export] Instance reflexive_proper_proxy {A : Type}
  {R : Relation A} `(Reflexive A R) (x : A)
  : CMorphisms.ProperProxy R x := reflexivity x.


(* forall (x y : A), x $== y -> forall (a b : A), a $== b -> y $== b -> x $==a *)

#[export] Instance IsProper_GpdHom_from {A : Type} `{Is0Gpd A}
  : CMorphisms.Proper
      (GpdHom ==>
         GpdHom ==>
         CRelationClasses.flip CRelationClasses.arrow) GpdHom.

Proof.

  intros x y eq_xy a b eq_ab eq_yb.

  exact (transitivity eq_xy (transitivity eq_yb
                              (symmetry _ _ eq_ab))).

Defined.


(* forall (x y : A), x $== y -> forall (a b : A), a $== b -> x $== a -> y $== b *)

#[export] Instance IsProper_GpdHom_to {A : Type} `{Is0Gpd A}
  : CMorphisms.Proper
      (GpdHom ==>
         GpdHom ==>
         CRelationClasses.arrow) GpdHom.

Proof.

  intros x y eq_xy a b eq_ab eq_yb.

  unshelve refine (transitivity _ eq_ab).

  unshelve refine (transitivity _ eq_yb).

  exact (symmetry _ _ eq_xy).

Defined.


(* forall a : A, x $== y -> a $== x -> a $== y *)

#[export] Instance IsProper_GpdHom_to_a {A : Type} `{Is0Gpd A}
  {a : A}
  : CMorphisms.Proper
      (GpdHom ==>
         CRelationClasses.arrow) (GpdHom a).

Proof.

  intros x y eq_xy eq_ax.

  now transitivity x.

Defined.


(* forall a : A, x $== y -> a $== y -> a $== x *)

#[export] Instance IsProper_GpdHom_from_a {A : Type} `{Is0Gpd A}
  {a : A}
  : CMorphisms.Proper
      (GpdHom ==>
         CRelationClasses.flip CRelationClasses.arrow) (GpdHom a).

Proof.

  intros x y eq_xy eq_ay.

  exact (transitivity eq_ay (symmetry _ _ eq_xy)).

Defined.


Open Scope signatureT_scope.


#[export] Instance symmetry_flip {A B : Type} {f : A -> B}
  {R : Relation A} {R' : Relation B} `{Symmetric _ R}
  (H0 : CMorphisms.Proper (R ++> R') f)
  : CMorphisms.Proper (R --> R') f.

Proof.

  intros a b Rab.

  apply H0.

 unfold CRelationClasses.flip.

 symmetry.

 exact Rab.

Defined.


#[export] Instance symmetric_flip_snd {A B C : Type} {R : Relation A}
  {R' : Relation B} {R'' : Relation C} `{Symmetric _ R'}
  (f : A -> B -> C) (H1 : CMorphisms.Proper (R ++> R' ++> R'') f)
  : CMorphisms.Proper (R ++> R' --> R'') f.

Proof.

  intros a b Rab x y R'yx.

 apply H1; [ assumption | symmetry; assumption ].

Defined.


#[export] Instance IsProper_fmap {A B : Type} `{Is1Cat A}
  `{Is1Cat A} (F : A -> B) `{Is1Functor _ _ F} (a b : A)
  : CMorphisms.Proper (GpdHom ==> GpdHom) (@fmap _ _ _ _ F _ a b) := fun _ _ eq => fmap2 F eq.


#[export] Instance IsProper_catcomp_g {A : Type} `{Is1Cat A}
  {a b c : A} (g : b $-> c)
  : CMorphisms.Proper (GpdHom ==> GpdHom) (@cat_comp _ _ _ a b c g).

Proof.

  intros f1 f2.

  apply (is0functor_postcomp a b c g ).

Defined.


#[export] Instance IsProper_catcomp {A : Type} `{Is1Cat A}
  {a b c : A}
  : CMorphisms.Proper (GpdHom ==> GpdHom ==> GpdHom)
      (@cat_comp _ _ _ a b c).

Proof.

  intros g1 g2 eq_g f1 f2 eq_f.

  rewrite eq_f.

  apply (is0functor_precomp a b c f2).

  exact eq_g.

Defined.


#[export] Instance gpd_hom_to_hom_proper {A B : Type} `{Is0Gpd A}
  {R : Relation B} (F : A -> B)
  `{CMorphisms.Proper _ (GpdHom ==> R) F}
  : CMorphisms.Proper (Hom ==> R) F.

Proof.

  intros a b eq_ab; apply H2; exact eq_ab.

Defined.


#[export] Instance gpd_hom_is_proper1 {A : Type} `{Is0Gpd A}
 : CMorphisms.Proper
     (Hom ==> Hom ==> CRelationClasses.arrow) Hom.

Proof.

  intros x y eq_xy a b eq_ab f.

  refine (transitivity _ eq_ab).

  refine (transitivity _ f).

  symmetry; exact eq_xy.

Defined.


#[export] Instance transitive_hom {A : Type} `{Is01Cat A} {x : A}
 : CMorphisms.Proper
     (Hom ==> CRelationClasses.arrow) (Hom x).

Proof.

  intros y z g f.

  exact (g $o f).

Defined.


Proposition IsEpic_HasSection {A} `{Is1Cat A}
  {a b : A} (f : a $-> b) :
  SectionOf f -> Epic f.

Proof.

  intros section c g h eq_gf_hf.

  destruct section as [right_inverse is_section].

  apply (is0functor_precomp _ _ _ right_inverse) in eq_gf_hf;
    unfold cat_precomp in eq_gf_hf.

  rewrite 2 cat_assoc, is_section, 2 cat_idr in eq_gf_hf.

  exact eq_gf_hf.

Defined.


Proposition IsMonic_HasRetraction {A} `{Is1Cat A}
  {b c : A} (f : b $-> c) :
  RetractionOf f -> Monic f.

Proof.

  intros retraction a g h eq_fg_fh.

  destruct retraction as [left_inverse is_retraction].

  apply (is0functor_postcomp _ _ _ left_inverse) in eq_fg_fh;
    unfold cat_postcomp in eq_fg_fh.

  rewrite <- 2 cat_assoc, is_retraction, 2 cat_idl in eq_fg_fh.

  assumption.

Defined.


Proposition nat_equiv_faithful {A B : Type}
  {F G : A -> B} `{Is1Functor _ _ F}
  `{!Is0Functor G, !Is1Functor G}
  `{!HasEquivs B} (tau : NatEquiv F G)
  : Faithful F -> Faithful G.

Proof.

  intros faithful_F x y f g eq_Gf_Gg.

  apply (@fmap _ _ _ _ _ (is0functor_precomp _
       _ _ (cat_equiv_natequiv tau x))) in eq_Gf_Gg.

  cbn in eq_Gf_Gg.

  unfold cat_precomp in eq_Gf_Gg.

  rewrite <- is1natural_natequiv in eq_Gf_Gg.

  rewrite <- is1natural_natequiv in eq_Gf_Gg.

  apply faithful_F.

  assert (X : RetractionOf (tau y)).

    unshelve eapply Build_RetractionOf.

 exact ((tau y)^-1$).

 exact (cate_issect _ ).

  apply IsMonic_HasRetraction in X.

  apply X in eq_Gf_Gg.

 assumption.

Defined.


Section SetoidRewriteTests.

  Goal forall (A : Type) `(H : Is0Gpd A) (a b c : A),
      a $== b -> b $== c -> a $== c.

  Proof.

    intros A ? ? ? a b c eq_ab eq_bc.

    rewrite eq_ab, <- eq_bc.

  Abort.

  Goal forall (A : Type) `(H : Is0Gpd A) (a b c : A),
      a $== b -> b $== c -> a $== c.

  Proof.

    intros A ? ? ? a b c eq_ab eq_bc.

    symmetry.

    rewrite eq_ab, <- eq_bc.

    rewrite eq_bc.

    rewrite <- eq_bc.

  Abort.


  Goal forall (A B : Type) (F : A -> B) `{Is1Functor _ _ F} (a b : A) (f g : a $-> b), f $== g -> fmap F f $== fmap F g.

  Proof.

    do 17 intro.

    intro eq_fg.

    rewrite eq_fg.

  Abort.


  Goal forall (A : Type) `{Is1Cat A} (a b c : A) (f1 f2 : a $-> b) (g : b $-> c), f1 $== f2 -> g $o f1 $== g $o f2.

  Proof.

    do 11 intro.

    intro eq.

    rewrite <- eq.

    rewrite eq.

  Abort.


  Goal forall (A : Type) `{Is1Cat A} (a b c : A) (f : a $-> b) (g1 g2 : b $-> c), g1 $== g2 -> g1 $o f $== g2 $o f.

  Proof.

  do 11 intro.

  intro eq.

  rewrite <- eq.

  rewrite eq.

  rewrite <- eq.

  Abort.


  Goal forall (A : Type) `{Is1Cat A} (a b c : A) (f1 f2 : a $-> b) (g1 g2 : b $-> c), g1 $== g2 -> f1 $== f2 -> g1 $o f1 $== g2 $o f2.

  Proof.

    do 12 intro.

    intros eq_g eq_f.

    rewrite eq_g.

    rewrite <- eq_f.

    rewrite eq_f.

    rewrite <- eq_g.

  Abort.

End SetoidRewriteTests.

Timings for SetoidRewrite.v