From Tealeaves Require Export
  Functors.Compose
  Classes.Functor2
  Classes.Categorical.Applicative
  Classes.Categorical.TraversableFunctor.

Import Functor2.Notations.

#[local] Generalizable Variable T G ϕ A B.
#[local] Arguments map F%function_scope {Map} {A B}%type_scope f%function_scope _.

Class ApplicativeDist2 (T: Type → Type → Type) :=
  dist2: ∀ (G: Type → Type)
           `{Map_G: Map G} `{Pure_G: Pure G} `{Mult_G: Mult G},
      T ○21 G ⇒2 G ○12 T.

#[global] Arguments dist2 {T}%function_scope {ApplicativeDist2}
  {G}%function_scope {Map_G Pure_G Mult_G} {B A}%type_scope.

#[local] Arguments dist2 T%function_scope {ApplicativeDist2}
  G%function_scope {Map_G Pure_G Mult_G} {B A}%type_scope.

Class TraversableFunctor2
  (T: Type → Type → Type)
  `{Map2_F: Map2 T}
  `{dist2_F: ApplicativeDist2 T} :=
  { trav2_functor :> Functor2 T;
    dist2_natural :> ∀ `{Applicative G},
        @Natural2 (T ○21 G) _ (G ○12 T) _ (@dist2 T dist2_F G _ _ _);
    dist2_morph: ∀ `{ApplicativeMorphism G1 G2 ϕ},
      `(dist2 T G2 ∘ map2 (ϕ B) (ϕ A) = ϕ (T B A) ∘ dist2 T G1);
    dist2_unit:
    `(dist2 T (fun A ⇒ A) = id (A := T B A));
    dist2_linear: ∀ `{Applicative G1} `{Applicative G2},
      `(dist2 T (G1 ∘ G2) = map G1 (dist2 T G2) ∘
                              dist2 T G1 (B := G2 B) (A := G2 A));
  }.

Section purity_law.

  Context
    `{TraversableFunctor2 T}.

(*
  Corollary map_purity_1: forall `{Applicative G},
  `(dist2 T G ∘ map2 T ((pure G) = pure G (A := T A)).
  Proof.
  intros. rewrite (dist_morph (ϕ := @pure G _)).
  now rewrite (dist_unit).
  Qed.

  Corollary map_purity_2:
  forall `{Applicative G1} `{Applicative G2} `(f: A -> G1 B),
  dist T (G2 ∘ G1) ∘ map T (pure G2 ∘ f) = pure G2 ∘ dist T G1 ∘ map T f.
  Proof.
  intros. rewrite <- (fun_map_map).
  reassociate <-. rewrite dist_linear.
  reassociate -> near (map T (pure G2)).
  rewrite map_purity_1.
  fequal. ext t. unfold compose.
  now rewrite app_pure_natural.
  Qed.
 *)

End purity_law.

Module Notations.
  Notation "'δ2'" := dist2: tealeaves_scope.
End Notations.