From Tealeaves Require Export
  Categories.TypeFamily
  Classes.Monoid
  Classes.Functor
  Classes.Categorical.Applicative
  Classes.Multisorted.Multifunctor
  Functors.Writer.

Import Categorical.Monad (Return, ret).

Import TypeFamily.Notations.
Import Product.Notations.
Import Monoid.Notations.

#[local] Generalizable Variables A B C F G ϕ W T U.

(** * Multisorted DTMs *)
(**********************************************************************)
Section MultisortedDTM_typeclasses.

  Context
    `{ix: Index}.

  (** ** Operations *)
  (********************************************************************)
  Section operations.

    Context
      (W: Type)
      (T: K -> Type -> Type)
      (U: Type -> Type).

    Class MReturn :=
      mret: forall (A: Type) (k: K), A -> T k A.

    Class MBind :=
      mbinddt: forall (F: Type -> Type) `{Map F} `{Pure F} `{Mult F} {A B: Type},
          (forall (k: K), W * A -> F (T k B)) -> U A -> F (U B).

  End operations.

  (** ** Kleisli Composition *)
  (********************************************************************)
  Definition compose_dtm
    {W: Type}
    {T: K -> Type -> Type}
    `{mn_op: Monoid_op W}
    `{mn_unit: Monoid_unit W}
    `{Map F} `{Mult F} `{Pure F}
    `{Map G} `{Mult G} `{Pure G}
    `{forall k, MBind W T (T k)}
    {A B C: Type}
    (g: forall k, W * B -> G (T k C))
    (f: forall k, W * A -> F (T k B)): forall k, W * A -> F (G (T k C)) :=
    fun (k: K) '(w, a) =>
      map (F := F)
        (mbinddt W T (T k) G (g ◻ allK (incr w))) (f k (w, a)).

  Infix "⋆dtm" := compose_dtm (at level 40): tealeaves_scope.

  (** ** Typeclasses *)
  (********************************************************************)

  (** *** PreModules *)
  (********************************************************************)
  Section PreModule.

    Context
      (W: Type)
      (T: K -> Type -> Type)
      (U: Type -> Type)
      `{! MReturn T}
      `{! MBind W T U}
      `{! forall k, MBind W T (T k)}
      {mn_op: Monoid_op W}
      {mn_unit: Monoid_unit W}.

    Class MultiDecoratedTraversablePreModule :=
      { dtp_monoid :> Monoid W;
        dtp_mbinddt_mret: forall A,
          mbinddt W T U (fun a => a) (mret T A ◻ allK extract) = @id (U A);
        dtp_mbinddt_mbinddt: forall
          (F: Type -> Type)
          (G: Type -> Type)
          `{Applicative F}
          `{Applicative G}
          `(g: forall k, W * B -> G (T k C))
          `(f: forall k, W * A -> F (T k B)),
          map (F := F) (mbinddt W T U G g) ∘ mbinddt W T U F f =
            mbinddt W T U (F ∘ G) (g ⋆dtm f);
        dtp_mbinddt_morphism: forall
          (F: Type -> Type)
          (G: Type -> Type)
          `{ApplicativeMorphism F G ϕ}
          `(f: forall k, W * A -> F (T k B)),
          ϕ (U B) ∘ mbinddt W T U F f =
            mbinddt W T U G ((fun k => ϕ (T k B)) ◻ f);
      }.

  End PreModule.

  (** *** DTMs *)
  (********************************************************************)
  Section DTM.

    Context
      (W: Type)
      (T: K -> Type -> Type)
      `{! MReturn T}
      `{! forall k, MBind W T (T k)}
      {mn_op: Monoid_op W}
      {mn_unit: Monoid_unit W}.

    Class MultiDecoratedTraversableMonad :=
      { dtm_pre :> forall k, MultiDecoratedTraversablePreModule W T (T k);
        dtm_mbinddt_comp_mret:
        forall k F `{Applicative F}
          `(f: forall k, W * A -> F (T k B)),
          mbinddt W T (T k) F f ∘ mret T A k = f k ∘ pair Ƶ;
      }.

  End DTM.

End MultisortedDTM_typeclasses.

Arguments mret {ix} _%function_scope {MReturn} {A}%type_scope _ _.
Arguments mbinddt {ix} {W}%type_scope {T} (U)%function_scope {MBind} F%function_scope {H H0 H1} {A B}.

#[local] Infix "⋆dtm" := compose_dtm (at level 40): tealeaves_scope.

(** ** Operation <<mapMret>> *)
(********************************************************************)
Section mapMret.

  Context
  `{ix: Index}
  `{! MReturn T}.

  Definition mapMret
    `{Map F} {A:Type}:
    forall (k: K), F A -> F (T k A) :=
    vec_apply (fun k => map (A := A) (B := T k A)) (mret T).

  Lemma vec_compose_mret {W A B}:
    forall (f: K -> W * A -> B) (k:K),
      (mret T ◻ f) k =
        (mret T k ∘ f k).
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
W, A, B: Type
forall (f : K -> W * A -> B) (k : K),
(mret T ◻ f) k = mret T k ∘ f k
  Proof.
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
W, A, B: Type
forall (f : K -> W * A -> B) (k : K),
(mret T ◻ f) k = mret T k ∘ f k
    reflexivity.
  Qed.

  Lemma vec_compose_mapMret {W A B} `{Functor F}:
    forall (f: K -> W * A -> F B) (k:K),
      (mapMret (F := F) ◻ f) k =
        (map (F := F) (mret T k) ∘ f k).
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
W, A, B: Type
F: Type -> Type
Map_F: Map F
H: Functor F
forall (f : K -> W * A -> F B) (k : K),
(mapMret ◻ f) k = map (mret T k) ∘ f k
  Proof.
ix: Index
T: K -> Type -> Type
MReturn0: MReturn T
W, A, B: Type
F: Type -> Type
Map_F: Map F
H: Functor F
forall (f : K -> W * A -> F B) (k : K),
(mapMret ◻ f) k = map (mret T k) ∘ f k
    reflexivity.
  Qed.

End mapMret.

(** ** Lemmas for Kleisli Composition *)
(********************************************************************)
Section multisorted_dtm_kleisli_composition.

  Context
    `{ix: Index}
    {W: Type}
    {T: K -> Type -> Type}
    {U: Type -> Type}
    `{! MReturn T}
    `{! MBind W T U}
    `{! forall k, MBind W T (T k)}
    {mn_op: Monoid_op W}
    {mn_unit: Monoid_unit W}.

  Context
    `{Applicative G}
    `{Applicative F}
    `{! Monoid W}
    {A B C: Type}
    {g: forall k, W * B -> G (T k C)}.

  Lemma compose_dtm_incr
    {f: forall k, W * A -> F (T k B)}:
      forall (w: W),
        (fun (k: K) => (g ⋆dtm f) k ∘ incr w) =
          ((fun (k: K) => g k ∘ incr w) ⋆dtm (fun (k: K) => f k ∘ incr w)).
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
forall w : W,
(fun k : K => (g ⋆dtm f) k ∘ incr w) =
(fun k : K => g k ∘ incr w)
⋆dtm (fun k : K => f k ∘ incr w)
  Proof.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
forall w : W,
(fun k : K => (g ⋆dtm f) k ∘ incr w) =
(fun k : K => g k ∘ incr w)
⋆dtm (fun k : K => f k ∘ incr w)
    intros.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
(fun k : K => (g ⋆dtm f) k ∘ incr w) =
(fun k : K => g k ∘ incr w)
⋆dtm (fun k : K => f k ∘ incr w) ext k [w' a].
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
((g ⋆dtm f) k ∘ incr w) (w', a) =
((fun k : K => g k ∘ incr w)
 ⋆dtm (fun k : K => f k ∘ incr w)) k (w', a)
    cbn.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
map (mbinddt (T k) G (g ◻ allK (incr (w ● w'))))
  (f k (w ● w', a)) =
map
  (mbinddt (T k) G
     ((fun k : K => g k ∘ incr w) ◻ allK (incr w')))
  ((f k ∘ incr w) (w', a)) do 2 fequal.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
g ◻ allK (incr (w ● w')) =
(fun k : K => g k ∘ incr w) ◻ allK (incr w')
    ext j [w'' b].
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
j: K
w'': W
b: B
(g ◻ allK (incr (w ● w'))) j (w'', b) =
((fun k : K => g k ∘ incr w) ◻ allK (incr w')) j
  (w'', b)
    unfold vec_compose, compose.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
j: K
w'': W
b: B
g j (allK (incr (w ● w')) j (w'', b)) =
g j (incr w (allK (incr w') j (w'', b))) cbn.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
j: K
w'': W
b: B
g j ((w ● w') ● w'', b) = g j (w ● (w' ● w''), b) fequal.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
j: K
w'': W
b: B
((w ● w') ● w'', b) = (w ● (w' ● w''), b)
    now rewrite monoid_assoc.
  Qed.

  Lemma compose_dtm_incr_alt
    {f: forall k, W * A -> F (T k B)}:
    forall (w: W),
      vec_compose
        (C := fun (k: K) => F (G (T k C)))
        (g ⋆dtm f) (allK (incr w)) =
        (g ◻ allK (incr w)) ⋆dtm (f ◻ allK (incr w)).
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
forall w : W,
g ⋆dtm f ◻ allK (incr w) =
(g ◻ allK (incr w)) ⋆dtm (f ◻ allK (incr w))
  Proof.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
forall w : W,
g ⋆dtm f ◻ allK (incr w) =
(g ◻ allK (incr w)) ⋆dtm (f ◻ allK (incr w))
    intros.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
g ⋆dtm f ◻ allK (incr w) =
(g ◻ allK (incr w)) ⋆dtm (f ◻ allK (incr w))
    ext k [w' a].
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
(g ⋆dtm f ◻ allK (incr w)) k (w', a) =
((g ◻ allK (incr w)) ⋆dtm (f ◻ allK (incr w))) k
  (w', a)
    cbn.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
map (mbinddt (T k) G (g ◻ allK (incr (w ● w'))))
  (f k (w ● w', a)) =
map
  (mbinddt (T k) G
     ((g ◻ allK (incr w)) ◻ allK (incr w')))
  ((f ◻ allK (incr w)) k (w', a)) do 2 fequal.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
g ◻ allK (incr (w ● w')) =
(g ◻ allK (incr w)) ◻ allK (incr w')
    ext j [w'' b].
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
j: K
w'': W
b: B
(g ◻ allK (incr (w ● w'))) j (w'', b) =
((g ◻ allK (incr w)) ◻ allK (incr w')) j (w'', b)
    unfold vec_compose, compose.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
j: K
w'': W
b: B
g j (allK (incr (w ● w')) j (w'', b)) =
g j (allK (incr w) j (allK (incr w') j (w'', b)))
    cbn.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
j: K
w'': W
b: B
g j ((w ● w') ● w'', b) = g j (w ● (w' ● w''), b) fequal.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
w: W
k: K
w': W
a: A
j: K
w'': W
b: B
((w ● w') ● w'', b) = (w ● (w' ● w''), b)
    now rewrite monoid_assoc.
  Qed.

  Context
    `{! MultiDecoratedTraversableMonad W T}.

  Lemma compose_dtm_lemma1
    {f: forall (k:K), W * A -> B}:
      g ⋆dtm (mret T ◻ f) =
        (fun (k: K) '(w, a) => g k (w, f k (w, a))).
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
g ⋆dtm (mret T ◻ f) =
(fun (k : K) '(w, a) => g k (w, f k (w, a)))
  Proof.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
g ⋆dtm (mret T ◻ f) =
(fun (k : K) '(w, a) => g k (w, f k (w, a)))
    intros.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
g ⋆dtm (mret T ◻ f) =
(fun (k : K) '(w, a) => g k (w, f k (w, a)))
    unfold compose_dtm.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
(fun (k : K) '(w, a) =>
 map (mbinddt (T k) G (g ◻ allK (incr w)))
   ((mret T ◻ f) k (w, a))) =
(fun (k : K) '(w, a) => g k (w, f k (w, a)))
    ext k [w a].
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
k: K
w: W
a: A
map (mbinddt (T k) G (g ◻ allK (incr w)))
  ((mret T ◻ f) k (w, a)) = g k (w, f k (w, a))
    unfold_ops @Map_I.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
k: K
w: W
a: A
mbinddt (T k) G (g ◻ allK (incr w))
  ((mret T ◻ f) k (w, a)) = g k (w, f k (w, a))
    rewrite vec_compose_mret.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
k: K
w: W
a: A
mbinddt (T k) G (g ◻ allK (incr w))
  ((mret T k ∘ f k) (w, a)) = g k (w, f k (w, a))
    unfold compose.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
k: K
w: W
a: A
mbinddt (T k) G (g ◻ allK (incr w))
  (mret T k (f k (w, a))) = g k (w, f k (w, a))
    compose near (f k (w, a)) on left.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
k: K
w: W
a: A
(mbinddt (T k) G (g ◻ allK (incr w)) ∘ mret T k)
  (f k (w, a)) = g k (w, f k (w, a))
    rewrite (dtm_mbinddt_comp_mret W T k); auto.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
k: K
w: W
a: A
((g ◻ allK (incr w)) k ∘ pair Ƶ) (f k (w, a)) =
g k (w, f k (w, a))
    rewrite vec_compose_k.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
k: K
w: W
a: A
(g k ∘ allK (incr w) k ∘ pair Ƶ) (f k (w, a)) =
g k (w, f k (w, a))
    reassociate -> on left.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
k: K
w: W
a: A
(g k ∘ (allK (incr w) k ∘ pair Ƶ)) (f k (w, a)) =
g k (w, f k (w, a))
    unfold allK, const.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
k: K
w: W
a: A
(g k ∘ (incr w ∘ pair Ƶ)) (f k (w, a)) =
g k (w, f k (w, a))
    rewrite pair_incr_zero.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> B
k: K
w: W
a: A
(g k ∘ pair w) (f k (w, a)) = g k (w, f k (w, a))
    reflexivity.
  Qed.

  Lemma compose_dtm_lemma2
    {f: forall (k:K), W * A -> F B}:
      g ⋆dtm (mapMret (F := F) ◻ f) =
        (fun (k: K) '(w, a) =>
           map (F := F) (g k ∘ pair w) (f k (w, a))).
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
g ⋆dtm (mapMret ◻ f) =
(fun (k : K) '(w, a) =>
 map (g k ∘ pair w) (f k (w, a)))
  Proof.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
g ⋆dtm (mapMret ◻ f) =
(fun (k : K) '(w, a) =>
 map (g k ∘ pair w) (f k (w, a)))
    intros.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
g ⋆dtm (mapMret ◻ f) =
(fun (k : K) '(w, a) =>
 map (g k ∘ pair w) (f k (w, a)))
    unfold compose_dtm.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
(fun (k : K) '(w, a) =>
 map (mbinddt (T k) G (g ◻ allK (incr w)))
   ((mapMret ◻ f) k (w, a))) =
(fun (k : K) '(w, a) =>
 map (g k ∘ pair w) (f k (w, a)))
    ext k [w a].
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
k: K
w: W
a: A
map (mbinddt (T k) G (g ◻ allK (incr w)))
  ((mapMret ◻ f) k (w, a)) =
map (g k ∘ pair w) (f k (w, a))
    rewrite vec_compose_mapMret.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
k: K
w: W
a: A
map (mbinddt (T k) G (g ◻ allK (incr w)))
  ((map (mret T k) ∘ f k) (w, a)) =
map (g k ∘ pair w) (f k (w, a))
    unfold compose.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
k: K
w: W
a: A
map (mbinddt (T k) G (g ◻ allK (incr w)))
  (map (mret T k) (f k (w, a))) =
map (g k ○ pair w) (f k (w, a))
    compose near (f k (w, a)) on left.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
k: K
w: W
a: A
(map (mbinddt (T k) G (g ◻ allK (incr w)))
 ∘ map (mret T k)) (f k (w, a)) =
map (g k ○ pair w) (f k (w, a))
    rewrite fun_map_map.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
k: K
w: W
a: A
map (mbinddt (T k) G (g ◻ allK (incr w)) ∘ mret T k)
  (f k (w, a)) = map (g k ○ pair w) (f k (w, a))
    rewrite (dtm_mbinddt_comp_mret W T k); auto.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
k: K
w: W
a: A
map ((g ◻ allK (incr w)) k ∘ pair Ƶ) (f k (w, a)) =
map (g k ○ pair w) (f k (w, a))
    rewrite vec_compose_k.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
k: K
w: W
a: A
map (g k ∘ allK (incr w) k ∘ pair Ƶ) (f k (w, a)) =
map (g k ○ pair w) (f k (w, a))
    reassociate -> on left.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
k: K
w: W
a: A
map (g k ∘ (allK (incr w) k ∘ pair Ƶ)) (f k (w, a)) =
map (g k ○ pair w) (f k (w, a))
    unfold allK, const.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
k: K
w: W
a: A
map (g k ∘ (incr w ∘ pair Ƶ)) (f k (w, a)) =
map (g k ○ pair w) (f k (w, a))
    rewrite pair_incr_zero.
ix: Index
W: Type
T: K -> Type -> Type
U: Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H1: Applicative F
Monoid0: Monoid W
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
f: K -> W * A -> F B
k: K
w: W
a: A
map (g k ∘ pair w) (f k (w, a)) =
map (g k ○ pair w) (f k (w, a))
    reflexivity.
  Qed.

End multisorted_dtm_kleisli_composition.

(** ** Purity Law *)
(**********************************************************************)
Section DTM_laws.

  Context
    (U: Type -> Type)
    `{MultiDecoratedTraversablePreModule W T U}
    `{! MultiDecoratedTraversableMonad W T}.

  Lemma mbinddt_pure: forall (A: Type) `{Applicative F},
      (let p := ((fun k => (@pure F _ (T k A))): forall k, T k A -> F (T k A)) in
       let r := (@mret ix T _ A) in
       let e := (allK extract): forall k, W * A -> A
       in mbinddt U F (p ◻ r ◻ e)) = pure (A := U A).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (F : Type -> Type) (Map_G : Map F)
  (Pure_G : Pure F) (Mult_G : Mult F),
Applicative F ->
(let p := fun k : K => pure in
 let r := mret T in
 let e := allK extract in mbinddt U F ((p ◻ r) ◻ e)) =
pure
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall (A : Type) (F : Type -> Type) (Map_G : Map F)
  (Pure_G : Pure F) (Mult_G : Mult F),
Applicative F ->
(let p := fun k : K => pure in
 let r := mret T in
 let e := allK extract in mbinddt U F ((p ◻ r) ◻ e)) =
pure
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
(let p := fun k : K => pure in
 let r := mret T in
 let e := allK extract in mbinddt U F ((p ◻ r) ◻ e)) =
pure
    cbn zeta.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
mbinddt U F
  (((fun k : K => pure) ◻ mret T) ◻ allK extract) =
pure
    rewrite vec_compose_assoc.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
mbinddt U F
  ((fun k : K => pure) ◻ (mret T ◻ allK extract)) =
pure
    rewrite <- (dtp_mbinddt_morphism W T U (fun x => x) F (ϕ := @pure F _)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
pure
∘ mbinddt U (fun x : Type => x)
    (mret T ◻ allK extract) = pure
    rewrite (dtp_mbinddt_mret W T U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
pure ∘ id = pure
    reflexivity.
  Qed.

End DTM_laws.

(** * Derived Instances *)
(**********************************************************************)
Section derived_operations.

  Context
    (U: Type -> Type)
    `{MultiDecoratedTraversablePreModule W T U}.

  (** ** Derived Multisorted Operations *)
  (********************************************************************)
  Section definitions.

    Context
      {A B: Type}
      (F: Type -> Type)
      `{Map F} `{Mult F} `{Pure F}.

    Definition mbindd (f: forall k, W * A -> T k B): U A -> U B :=
      mbinddt U (fun x => x) f.

    Definition mmapdt (f: forall (k: K), W * A -> F B): U A -> F (U B) :=
      mbinddt U F (mapMret (T := T) ◻ f).

    Definition mbindt (f: forall k, A -> F (T k B)): U A -> F (U B) :=
      mbinddt U F (f ◻ allK extract).

    Definition mbind (f: forall k, A -> T k B): U A -> U B :=
      mbindd (f ◻ allK extract).

    Definition mmapd (f: forall k, W * A -> B): U A -> U B :=
      mbindd (mret T ◻ f).

    Definition mmapt (f: forall k, A -> F B): U A -> F (U B) :=
      mmapdt (f ◻ allK extract).

    Definition mmap (f: forall k, A -> B): U A -> U B :=
      mmapd (f ◻ allK extract).

  End definitions.

  (** ** Rewriting Rules *)
  (********************************************************************)
  Section special_cases.

    Context
      {A B: Type}
      (F: Type -> Type)
      `{Map F} `{Mult F} `{Pure F}.

    (** *** Special cases of <<mbinddt>> *)
    (******************************************************************)
    Lemma mbindt_to_mbinddt (f: forall k, A -> F (T k B)):
      mbindt F f = mbinddt U F (f ◻ allK extract).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: forall k : K, A -> F (T k B)
mbindt F f = mbinddt U F (f ◻ allK extract)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: forall k : K, A -> F (T k B)
mbindt F f = mbinddt U F (f ◻ allK extract)
      reflexivity.
    Qed.

    Lemma mbindd_to_mbinddt (f: forall k, W * A -> T k B):
      mbindd f = mbinddt U (fun A => A) f.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: W * A ~k~> T B
mbindd f = mbinddt U (fun A : Type => A) f
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: W * A ~k~> T B
mbindd f = mbinddt U (fun A : Type => A) f
      reflexivity.
    Qed.

    Lemma mmapdt_to_mbinddt (f: K -> W * A -> F B):
      mmapdt F f = mbinddt U F (mapMret (T := T) ◻ f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> W * A -> F B
mmapdt F f = mbinddt U F (mapMret ◻ f)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> W * A -> F B
mmapdt F f = mbinddt U F (mapMret ◻ f)
      reflexivity.
    Qed.

    Lemma mbind_to_mbinddt (f: forall k, A -> T k B):
      mbind f = mbinddt U (fun A => A) (f ◻ allK extract).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: A ~k~> T B
mbind f =
mbinddt U (fun A : Type => A) (f ◻ allK extract)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: A ~k~> T B
mbind f =
mbinddt U (fun A : Type => A) (f ◻ allK extract)
      reflexivity.
    Qed.

    Lemma mmapd_to_mbinddt (f: K -> W * A -> B):
      mmapd f = mbinddt U (fun A => A) (mret T ◻ f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> W * A -> B
mmapd f = mbinddt U (fun A : Type => A) (mret T ◻ f)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> W * A -> B
mmapd f = mbinddt U (fun A : Type => A) (mret T ◻ f)
      reflexivity.
    Qed.

    Lemma mmapt_to_mbinddt (f: K -> A -> F B):
      mmapt F f = mbinddt U F (mapMret (T := T)
                                 ◻ f ◻ allK extract).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> F B
mmapt F f = mbinddt U F ((mapMret ◻ f) ◻ allK extract)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> F B
mmapt F f = mbinddt U F ((mapMret ◻ f) ◻ allK extract)
      reflexivity.
    Qed.

    Lemma mmap_to_mbinddt (f: K -> A -> B):
      mmap f = mbinddt U (fun A => A) (mret T ◻ f ◻ allK extract).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> B
mmap f =
mbinddt U (fun A : Type => A)
  ((mret T ◻ f) ◻ allK extract)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> B
mmap f =
mbinddt U (fun A : Type => A)
  ((mret T ◻ f) ◻ allK extract)
      reflexivity.
    Qed.

    (** *** Special Cases of <<mbindt>> *)
    (******************************************************************)
    Lemma mbind_to_mbindt (f: forall k, A -> T k B):
      mbind f = mbindt (fun A => A) f.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: A ~k~> T B
mbind f = mbindt (fun A : Type => A) f
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: A ~k~> T B
mbind f = mbindt (fun A : Type => A) f
      reflexivity.
    Qed.

    Lemma mmapt_to_mbindt (f: K -> A -> F B):
      mmapt F f = mbindt F (mapMret (T := T) ◻ f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> F B
mmapt F f = mbindt F (mapMret ◻ f)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> F B
mmapt F f = mbindt F (mapMret ◻ f)
      reflexivity.
    Qed.

    Lemma mmap_to_mbindt (f: K -> A -> B):
      mmap f = mbindt (fun A => A) (mret T ◻ f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> B
mmap f = mbindt (fun A : Type => A) (mret T ◻ f)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> B
mmap f = mbindt (fun A : Type => A) (mret T ◻ f)
      reflexivity.
    Qed.

    (** *** Special Cases of <<mbindd>> *)
    (******************************************************************)
    Lemma mbind_to_mbindd (f: forall k, A -> T k B):
      mbind f = mbindd (f ◻ allK extract).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: A ~k~> T B
mbind f = mbindd (f ◻ allK extract)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: A ~k~> T B
mbind f = mbindd (f ◻ allK extract)
      reflexivity.
    Qed.

    Lemma mmapd_to_mbindd (f: W * A -k-> B):
      mmapd f = mbindd (mret T ◻ f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: W * A -k-> B
mmapd f = mbindd (mret T ◻ f)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: W * A -k-> B
mmapd f = mbindd (mret T ◻ f)
      reflexivity.
    Qed.

    Lemma mmap_to_mbindd (f: A -k-> B):
      mmap f = mbindd (mret T ◻ f ◻ allK extract).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: A -k-> B
mmap f = mbindd ((mret T ◻ f) ◻ allK extract)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: A -k-> B
mmap f = mbindd ((mret T ◻ f) ◻ allK extract)
      reflexivity.
    Qed.

    (** *** Special Cases of <<mbindd>> *)
    (******************************************************************)
    Lemma mmapd_to_mmapdt (f: K -> W * A -> B):
      mmapd f = mmapdt (fun A => A) f.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> W * A -> B
mmapd f = mmapdt (fun A : Type => A) f
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> W * A -> B
mmapd f = mmapdt (fun A : Type => A) f
      reflexivity.
    Qed.

    Lemma mmap_to_mmapdt (f: K -> A -> B):
      mmap f = mmapdt (fun A => A) (f ◻ allK extract).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> B
mmap f = mmapdt (fun A : Type => A) (f ◻ allK extract)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> B
mmap f = mmapdt (fun A : Type => A) (f ◻ allK extract)
      reflexivity.
    Qed.

    Lemma mmapt_to_mmapdt (f: K -> A -> F B):
      mmapt F f = mmapdt F (f ◻ allK extract).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> F B
mmapt F f = mmapdt F (f ◻ allK extract)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> F B
mmapt F f = mmapdt F (f ◻ allK extract)
      reflexivity.
    Qed.

    (** *** Special Cases of <<mmapt>> *)
    (******************************************************************)
    Lemma mmap_to_mmapt (f: K -> A -> B):
      mmap f = mmapt (fun A => A) f.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> B
mmap f = mmapt (fun A : Type => A) f
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> B
mmap f = mmapt (fun A : Type => A) f
      reflexivity.
    Qed.

    (** *** Special Cases of <<mmapd>> *)
    (******************************************************************)
    Lemma mmap_to_mmapd (f: K -> A -> B):
      mmap f = mmapd (f ◻ allK extract).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> B
mmap f = mmapd (f ◻ allK extract)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> B
mmap f = mmapd (f ◻ allK extract)
      reflexivity.
    Qed.

    (** *** Special Cases of <<mbind>> *)
    (******************************************************************)
    Lemma mmap_to_mbind (f: K -> A -> B):
      mmap f = mbind (mret T ◻ f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> B
mmap f = mbind (mret T ◻ f)
    Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
A, B: Type
F: Type -> Type
H1: Map F
H2: Mult F
H3: Pure F
f: K -> A -> B
mmap f = mbind (mret T ◻ f)
      reflexivity.
    Qed.

  End special_cases.

End derived_operations.

(** ** Composition Between <<mbinddt>> and Other Operations *)
(** Compositions laws for compositions of the form <<mbinddt ∘ xxx>> or
    <<xxx ∘ mbinddt>> *)
(**********************************************************************)
Section derived_operations_composition.

  Context
    (U: Type -> Type)
    `{MultiDecoratedTraversablePreModule W T U}
    `{! MultiDecoratedTraversableMonad W T}
    `{Applicative F}
    `{Applicative G}
    {A B C: Type}.

  (** *** <<mbinddt>> on the right *)
  (********************************************************************)
  Lemma mbindd_mbinddt: forall
      (g: forall k, W * B -> T k C)
      (f: forall k, W * A -> F (T k B)),
      map (F := F) (mbindd U g) ∘ mbinddt U F f =
        mbinddt U F
          (fun k '(w, a) =>
             map (F := F)
               (mbindd (T k) (g ◻ allK (incr w))) (f k (w, a))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : W * B ~k~> T C)
  (f : forall k : K, W * A -> F (T k B)),
map (mbindd U g) ∘ mbinddt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mbindd (T k) (g ◻ allK (incr w))) (f k (w, a)))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : W * B ~k~> T C)
  (f : forall k : K, W * A -> F (T k B)),
map (mbindd U g) ∘ mbinddt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mbindd (T k) (g ◻ allK (incr w))) (f k (w, a)))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: W * B ~k~> T C
f: forall k : K, W * A -> F (T k B)
map (mbindd U g) ∘ mbinddt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mbindd (T k) (g ◻ allK (incr w))) (f k (w, a))) rewrite mbindd_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: W * B ~k~> T C
f: forall k : K, W * A -> F (T k B)
map (mbinddt U (fun A : Type => A) g) ∘ mbinddt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mbindd (T k) (g ◻ allK (incr w))) (f k (w, a)))
    rewrite (dtp_mbinddt_mbinddt W T U F (fun A => A)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: W * B ~k~> T C
f: forall k : K, W * A -> F (T k B)
mbinddt U (F ∘ (fun A : Type => A)) (g ⋆dtm f) =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mbindd (T k) (g ◻ allK (incr w))) (f k (w, a)))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: W * B ~k~> T C
f: forall k : K, W * A -> F (T k B)
Mult_compose F (fun A : Type => A) = Mult_G now erewrite Mult_compose_identity1.
  Qed.

  Lemma mmapdt_mbinddt: forall
      (g: K -> W * B -> G C)
      (f: forall k, W * A -> F (T k B)),
      map (F := F) (mmapdt U G g) ∘ mbinddt U F f =
        mbinddt U (F ∘ G)
          (fun k '(w, a) =>
             map (F := F)
               (mmapdt (T k) G (g ◻ allK (incr w))) (f k (w, a))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : K -> W * B -> G C)
  (f : forall k : K, W * A -> F (T k B)),
map (mmapdt U G g) ∘ mbinddt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (mmapdt (T k) G (g ◻ allK (incr w)))
     (f k (w, a)))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : K -> W * B -> G C)
  (f : forall k : K, W * A -> F (T k B)),
map (mmapdt U G g) ∘ mbinddt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (mmapdt (T k) G (g ◻ allK (incr w)))
     (f k (w, a)))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> W * B -> G C
f: forall k : K, W * A -> F (T k B)
map (mmapdt U G g) ∘ mbinddt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (mmapdt (T k) G (g ◻ allK (incr w)))
     (f k (w, a))) rewrite mmapdt_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> W * B -> G C
f: forall k : K, W * A -> F (T k B)
map (mbinddt U G (mapMret ◻ g)) ∘ mbinddt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (mmapdt (T k) G (g ◻ allK (incr w)))
     (f k (w, a)))
    now rewrite (dtp_mbinddt_mbinddt W T U F G).
  Qed.

  Lemma mbindt_mbinddt: forall
      (g: forall k, B -> G (T k C))
      (f: forall k, W * A -> F (T k B)),
      map (F := F) (mbindt U G g) ∘ mbinddt U F f =
        mbinddt U (F ∘ G)
          (fun k => map (F := F) (mbindt (T k) G g) ∘ f k).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, B -> G (T k C))
  (f : forall k : K, W * A -> F (T k B)),
map (mbindt U G g) ∘ mbinddt U F f =
mbinddt U (F ∘ G)
  (fun k : K => map (mbindt (T k) G g) ∘ f k)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, B -> G (T k C))
  (f : forall k : K, W * A -> F (T k B)),
map (mbindt U G g) ∘ mbinddt U F f =
mbinddt U (F ∘ G)
  (fun k : K => map (mbindt (T k) G g) ∘ f k)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
map (mbindt U G g) ∘ mbinddt U F f =
mbinddt U (F ∘ G)
  (fun k : K => map (mbindt (T k) G g) ∘ f k) rewrite mbindt_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
map (mbinddt U G (g ◻ allK extract)) ∘ mbinddt U F f =
mbinddt U (F ∘ G)
  (fun k : K => map (mbindt (T k) G g) ∘ f k)
    rewrite (dtp_mbinddt_mbinddt W T U F G).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
mbinddt U (F ∘ G) ((g ◻ allK extract) ⋆dtm f) =
mbinddt U (F ∘ G)
  (fun k : K => map (mbindt (T k) G g) ∘ f k)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
(g ◻ allK extract) ⋆dtm f =
(fun k : K => map (mbindt (T k) G g) ∘ f k) ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
((g ◻ allK extract) ⋆dtm f) k (w, a) =
(map (mbindt (T k) G g) ∘ f k) (w, a) unfold compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
map
  (mbinddt (T k) G
     ((g ◻ allK extract) ◻ allK (incr w)))
  (f k (w, a)) = map (mbindt (T k) G g) (f k (w, a))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
mbinddt (T k) G ((g ◻ allK extract) ◻ allK (incr w)) =
mbindt (T k) G g rewrite mbindt_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
mbinddt (T k) G ((g ◻ allK extract) ◻ allK (incr w)) =
mbinddt (T k) G (g ◻ allK extract)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, B -> G (T k C)
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
(g ◻ allK extract) ◻ allK (incr w) = g ◻ allK extract now ext j [w2 b].
  Qed.

  Lemma mbind_mbinddt: forall
      (g: forall k, B -> T k C)
      (f: forall k, W * A -> F (T k B)),
      map (F := F) (mbind U g) ∘ mbinddt U F f =
        mbinddt U F ((fun k => map (F := F) (mbind (T k) g)) ◻ f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : B ~k~> T C)
  (f : forall k : K, W * A -> F (T k B)),
map (mbind U g) ∘ mbinddt U F f =
mbinddt U F ((fun k : K => map (mbind (T k) g)) ◻ f)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : B ~k~> T C)
  (f : forall k : K, W * A -> F (T k B)),
map (mbind U g) ∘ mbinddt U F f =
mbinddt U F ((fun k : K => map (mbind (T k) g)) ◻ f)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: B ~k~> T C
f: forall k : K, W * A -> F (T k B)
map (mbind U g) ∘ mbinddt U F f =
mbinddt U F ((fun k : K => map (mbind (T k) g)) ◻ f) rewrite mbind_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: B ~k~> T C
f: forall k : K, W * A -> F (T k B)
map (mbinddt U (fun A : Type => A) (g ◻ allK extract))
∘ mbinddt U F f =
mbinddt U F ((fun k : K => map (mbind (T k) g)) ◻ f)
    rewrite (dtp_mbinddt_mbinddt W T U F (fun A => A)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: B ~k~> T C
f: forall k : K, W * A -> F (T k B)
mbinddt U (F ∘ (fun A : Type => A))
  ((g ◻ allK extract) ⋆dtm f) =
mbinddt U F ((fun k : K => map (mbind (T k) g)) ◻ f)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: B ~k~> T C
f: forall k : K, W * A -> F (T k B)
Mult_compose F (fun A : Type => A) = Mult_G
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: B ~k~> T C
f: forall k : K, W * A -> F (T k B)
(g ◻ allK extract) ⋆dtm f =
(fun k : K => map (mbind (T k) g)) ◻ f
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: B ~k~> T C
f: forall k : K, W * A -> F (T k B)
Mult_compose F (fun A : Type => A) = Mult_G now erewrite Mult_compose_identity1.
    -
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: B ~k~> T C
f: forall k : K, W * A -> F (T k B)
(g ◻ allK extract) ⋆dtm f =
(fun k : K => map (mbind (T k) g)) ◻ f unfold vec_compose, compose, compose_dtm.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: B ~k~> T C
f: forall k : K, W * A -> F (T k B)
(fun (k : K) '(w, a) =>
 map
   (mbinddt (T k) (fun A : Type => A)
      ((fun k0 : K => g k0 ○ allK extract k0)
       ◻ allK (incr w))) (f k (w, a))) =
(fun k : K => map (mbind (T k) g) ○ f k)
      ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: B ~k~> T C
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
map
  (mbinddt (T k) (fun A : Type => A)
     ((fun k : K => g k ○ allK extract k)
      ◻ allK (incr w))) (f k (w, a)) =
map (mbind (T k) g) (f k (w, a))
      fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: B ~k~> T C
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
mbinddt (T k) (fun A : Type => A)
  ((fun k : K => g k ○ allK extract k) ◻ allK (incr w)) =
mbind (T k) g rewrite mbind_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: B ~k~> T C
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
mbinddt (T k) (fun A : Type => A)
  ((fun k : K => g k ○ allK extract k) ◻ allK (incr w)) =
mbinddt (T k) (fun A : Type => A) (g ◻ allK extract) fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: B ~k~> T C
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
(fun k : K => g k ○ allK extract k) ◻ allK (incr w) =
g ◻ allK extract
      now ext j [w2 b].
  Qed.

  Lemma mmapd_mbinddt: forall
      (g: K -> W * B -> C)
      (f: forall k, W * A -> F (T k B)),
      map (F := F) (mmapd U g) ∘ mbinddt U F f =
        mbinddt U F
          (fun k '(w, a) =>
             map (F := F)
               (mmapd (T k) (g ◻ allK (incr w))) (f k (w, a))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : K -> W * B -> C)
  (f : forall k : K, W * A -> F (T k B)),
map (mmapd U g) ∘ mbinddt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mmapd (T k) (g ◻ allK (incr w))) (f k (w, a)))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : K -> W * B -> C)
  (f : forall k : K, W * A -> F (T k B)),
map (mmapd U g) ∘ mbinddt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mmapd (T k) (g ◻ allK (incr w))) (f k (w, a)))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> W * B -> C
f: forall k : K, W * A -> F (T k B)
map (mmapd U g) ∘ mbinddt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mmapd (T k) (g ◻ allK (incr w))) (f k (w, a))) rewrite mmapd_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> W * B -> C
f: forall k : K, W * A -> F (T k B)
map (mbinddt U (fun A : Type => A) (mret T ◻ g))
∘ mbinddt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mmapd (T k) (g ◻ allK (incr w))) (f k (w, a)))
    rewrite (dtp_mbinddt_mbinddt W T U F (fun A => A)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> W * B -> C
f: forall k : K, W * A -> F (T k B)
mbinddt U (F ∘ (fun A : Type => A))
  ((mret T ◻ g) ⋆dtm f) =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mmapd (T k) (g ◻ allK (incr w))) (f k (w, a)))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> W * B -> C
f: forall k : K, W * A -> F (T k B)
Mult_compose F (fun A : Type => A) = Mult_G now erewrite Mult_compose_identity1.
  Qed.

  Lemma mmapt_mbinddt: forall
      (g: K -> B -> G C)
      (f: forall k, W * A -> F (T k B)),
      map (F := F) (mmapt U G g) ∘ mbinddt U F f =
        mbinddt U (F ∘ G)
          (fun k => map (F := F) (mmapt (T k) G g) ∘ f k).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : K -> B -> G C)
  (f : forall k : K, W * A -> F (T k B)),
map (mmapt U G g) ∘ mbinddt U F f =
mbinddt U (F ∘ G)
  (fun k : K => map (mmapt (T k) G g) ∘ f k)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : K -> B -> G C)
  (f : forall k : K, W * A -> F (T k B)),
map (mmapt U G g) ∘ mbinddt U F f =
mbinddt U (F ∘ G)
  (fun k : K => map (mmapt (T k) G g) ∘ f k)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> G C
f: forall k : K, W * A -> F (T k B)
map (mmapt U G g) ∘ mbinddt U F f =
mbinddt U (F ∘ G)
  (fun k : K => map (mmapt (T k) G g) ∘ f k) rewrite mmapt_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> G C
f: forall k : K, W * A -> F (T k B)
map (mbinddt U G ((mapMret ◻ g) ◻ allK extract))
∘ mbinddt U F f =
mbinddt U (F ∘ G)
  (fun k : K => map (mmapt (T k) G g) ∘ f k)
    rewrite (dtp_mbinddt_mbinddt W T U F G).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> G C
f: forall k : K, W * A -> F (T k B)
mbinddt U (F ∘ G)
  (((mapMret ◻ g) ◻ allK extract) ⋆dtm f) =
mbinddt U (F ∘ G)
  (fun k : K => map (mmapt (T k) G g) ∘ f k)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> G C
f: forall k : K, W * A -> F (T k B)
((mapMret ◻ g) ◻ allK extract) ⋆dtm f =
(fun k : K => map (mmapt (T k) G g) ∘ f k) ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> G C
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
(((mapMret ◻ g) ◻ allK extract) ⋆dtm f) k (w, a) =
(map (mmapt (T k) G g) ∘ f k) (w, a) unfold compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> G C
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
map
  (mbinddt (T k) G
     (((mapMret ◻ g) ◻ allK extract) ◻ allK (incr w)))
  (f k (w, a)) = map (mmapt (T k) G g) (f k (w, a))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> G C
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
mbinddt (T k) G
  (((mapMret ◻ g) ◻ allK extract) ◻ allK (incr w)) =
mmapt (T k) G g rewrite mmapt_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> G C
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
mbinddt (T k) G
  (((mapMret ◻ g) ◻ allK extract) ◻ allK (incr w)) =
mbinddt (T k) G ((mapMret ◻ g) ◻ allK extract)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> G C
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
((mapMret ◻ g) ◻ allK extract) ◻ allK (incr w) =
(mapMret ◻ g) ◻ allK extract now ext j [w2 b].
  Qed.

  Lemma mmap_mbinddt: forall
      (g: K -> B -> C)
      (f: forall k, W * A -> F (T k B)),
      map (F := F) (mmap U g) ∘ mbinddt U F f =
        mbinddt U F (fun k => map (F := F) (mmap (T k) g) ∘ f k).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : K -> B -> C)
  (f : forall k : K, W * A -> F (T k B)),
map (mmap U g) ∘ mbinddt U F f =
mbinddt U F (fun k : K => map (mmap (T k) g) ∘ f k)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : K -> B -> C)
  (f : forall k : K, W * A -> F (T k B)),
map (mmap U g) ∘ mbinddt U F f =
mbinddt U F (fun k : K => map (mmap (T k) g) ∘ f k)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> C
f: forall k : K, W * A -> F (T k B)
map (mmap U g) ∘ mbinddt U F f =
mbinddt U F (fun k : K => map (mmap (T k) g) ∘ f k) unfold mmap.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> C
f: forall k : K, W * A -> F (T k B)
map (mmapd U (g ◻ allK extract)) ∘ mbinddt U F f =
mbinddt U F
  (fun k : K =>
   map (mmapd (T k) (g ◻ allK extract)) ∘ f k) rewrite mmapd_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> C
f: forall k : K, W * A -> F (T k B)
mbinddt U F
  (fun (k : K) '(w, a) =>
   map
     (mmapd (T k) ((g ◻ allK extract) ◻ allK (incr w)))
     (f k (w, a))) =
mbinddt U F
  (fun k : K =>
   map (mmapd (T k) (g ◻ allK extract)) ∘ f k)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> C
f: forall k : K, W * A -> F (T k B)
(fun (k : K) '(w, a) =>
 map
   (mmapd (T k) ((g ◻ allK extract) ◻ allK (incr w)))
   (f k (w, a))) =
(fun k : K =>
 map (mmapd (T k) (g ◻ allK extract)) ∘ f k) ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> C
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
map (mmapd (T k) ((g ◻ allK extract) ◻ allK (incr w)))
  (f k (w, a)) =
(map (mmapd (T k) (g ◻ allK extract)) ∘ f k) (w, a) unfold compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> C
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
map (mmapd (T k) ((g ◻ allK extract) ◻ allK (incr w)))
  (f k (w, a)) =
map (mmapd (T k) (g ◻ allK extract)) (f k (w, a))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> C
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
mmapd (T k) ((g ◻ allK extract) ◻ allK (incr w)) =
mmapd (T k) (g ◻ allK extract) fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: K -> B -> C
f: forall k : K, W * A -> F (T k B)
k: K
w: W
a: A
(g ◻ allK extract) ◻ allK (incr w) = g ◻ allK extract now ext j [w2 b].
  Qed.

  (** *** <<mbinddt>> on the left *)
  (********************************************************************)
  Lemma mbinddt_mbindd: forall
      (g: forall k, W * B -> G (T k C))
      (f: forall k, W * A -> T k B),
      mbinddt U G g ∘ mbindd U f =
        mbinddt U G
          (fun k '(w, a) =>
             mbinddt (T k) G (g ◻ allK (incr w)) (f k (w, a))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : W * A ~k~> T B),
mbinddt U G g ∘ mbindd U f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbinddt (T k) G (g ◻ allK (incr w)) (f k (w, a)))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : W * A ~k~> T B),
mbinddt U G g ∘ mbindd U f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbinddt (T k) G (g ◻ allK (incr w)) (f k (w, a)))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: W * A ~k~> T B
mbinddt U G g ∘ mbindd U f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbinddt (T k) G (g ◻ allK (incr w)) (f k (w, a))) rewrite mbindd_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: W * A ~k~> T B
mbinddt U G g ∘ mbinddt U (fun A : Type => A) f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbinddt (T k) G (g ◻ allK (incr w)) (f k (w, a)))
    change (mbinddt U G g) with
      (map (F := (fun A => A)) (mbinddt U G g)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: W * A ~k~> T B
map (mbinddt U G g) ∘ mbinddt U (fun A : Type => A) f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbinddt (T k) G (g ◻ allK (incr w)) (f k (w, a)))
    rewrite (dtp_mbinddt_mbinddt W T U (fun A => A) G).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: W * A ~k~> T B
mbinddt U ((fun A : Type => A) ∘ G) (g ⋆dtm f) =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbinddt (T k) G (g ◻ allK (incr w)) (f k (w, a)))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: W * A ~k~> T B
Mult_compose (fun A : Type => A) G = Mult_G0 now rewrite (Mult_compose_identity2 G).
  Qed.


  Lemma mbinddt_mmapdt: forall
      (g: forall k, W * B -> G (T k C))
      (f: K -> W * A -> F B),
      map (F := F) (mbinddt U G g) ∘ mmapdt U F f =
        mbinddt U (F ∘ G)
          (fun k '(w, a) => map (F := F) (fun b => g k (w, b)) (f k (w, a))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : K -> W * A -> F B),
map (mbinddt U G g) ∘ mmapdt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : K -> W * A -> F B),
map (mbinddt U G g) ∘ mmapdt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
map (mbinddt U G g) ∘ mmapdt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
    rewrite mmapdt_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
map (mbinddt U G g) ∘ mbinddt U F (mapMret ◻ f) =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
    rewrite (dtp_mbinddt_mbinddt W T U F G).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
mbinddt U (F ∘ G) (g ⋆dtm (mapMret ◻ f)) =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
g ⋆dtm (mapMret ◻ f) =
(fun (k : K) '(w, a) =>
 map (g k ○ pair w) (f k (w, a)))
    ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
k: K
w: W
a: A
(g ⋆dtm (mapMret ◻ f)) k (w, a) =
map (g k ○ pair w) (f k (w, a)) unfold compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
k: K
w: W
a: A
map (mbinddt (T k) G (g ◻ allK (incr w)))
  ((mapMret ◻ f) k (w, a)) =
map (g k ○ pair w) (f k (w, a))
    rewrite vec_compose_mapMret.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
k: K
w: W
a: A
map (mbinddt (T k) G (g ◻ allK (incr w)))
  ((map (mret T k) ∘ f k) (w, a)) =
map (g k ○ pair w) (f k (w, a))
    unfold compose at 1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
k: K
w: W
a: A
map (mbinddt (T k) G (g ◻ allK (incr w)))
  (map (mret T k) (f k (w, a))) =
map (g k ○ pair w) (f k (w, a))
    compose near (f k (w, a)) on left.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
k: K
w: W
a: A
(map (mbinddt (T k) G (g ◻ allK (incr w)))
 ∘ map (mret T k)) (f k (w, a)) =
map (g k ○ pair w) (f k (w, a))
    rewrite fun_map_map.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
k: K
w: W
a: A
map (mbinddt (T k) G (g ◻ allK (incr w)) ∘ mret T k)
  (f k (w, a)) = map (g k ○ pair w) (f k (w, a))
    rewrite (dtm_mbinddt_comp_mret W T k G).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
k: K
w: W
a: A
map ((g ◻ allK (incr w)) k ∘ pair Ƶ) (f k (w, a)) =
map (g k ○ pair w) (f k (w, a))
    rewrite vec_compose_allK2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
k: K
w: W
a: A
map (g k ∘ incr w ∘ pair Ƶ) (f k (w, a)) =
map (g k ○ pair w) (f k (w, a))
    reassociate -> on left.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
k: K
w: W
a: A
map (g k ∘ (incr w ∘ pair Ƶ)) (f k (w, a)) =
map (g k ○ pair w) (f k (w, a))
    rewrite pair_incr_zero.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> F B
k: K
w: W
a: A
map (g k ∘ pair w) (f k (w, a)) =
map (g k ○ pair w) (f k (w, a))
    reflexivity.
  Qed.

  Lemma mbinddt_mbindt: forall
      (g: forall k, W * B -> G (T k C))
      (f: forall k, A -> F (T k B)),
      map (F := F) (mbinddt U G g) ∘ mbindt U F f =
        mbinddt U (F ∘ G)
          (fun k '(w, a) =>
             map (F := F)
               (mbinddt (T k) G (g ◻ allK (incr w))) (f k a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : forall k : K, A -> F (T k B)),
map (mbinddt U G g) ∘ mbindt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (mbinddt (T k) G (g ◻ allK (incr w))) (f k a))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : forall k : K, A -> F (T k B)),
map (mbinddt U G g) ∘ mbindt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (mbinddt (T k) G (g ◻ allK (incr w))) (f k a))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, A -> F (T k B)
map (mbinddt U G g) ∘ mbindt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (mbinddt (T k) G (g ◻ allK (incr w))) (f k a))
    rewrite mbindt_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, A -> F (T k B)
map (mbinddt U G g) ∘ mbinddt U F (f ◻ allK extract) =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (mbinddt (T k) G (g ◻ allK (incr w))) (f k a))
    rewrite (dtp_mbinddt_mbinddt W T U F G).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: forall k : K, A -> F (T k B)
mbinddt U (F ∘ G) (g ⋆dtm (f ◻ allK extract)) =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (mbinddt (T k) G (g ◻ allK (incr w))) (f k a))
    reflexivity.
  Qed.

  Lemma mbinddt_mbind: forall
      (g: forall k, W * B -> G (T k C))
      (f: forall k, A -> T k B),
      mbinddt U G g ∘ mbind U f =
        mbinddt U G
          (fun k '(w, a) =>
             mbinddt (T k) G (g ◻ allK (incr w)) (f k a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : A ~k~> T B),
mbinddt U G g ∘ mbind U f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbinddt (T k) G (g ◻ allK (incr w)) (f k a))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : A ~k~> T B),
mbinddt U G g ∘ mbind U f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbinddt (T k) G (g ◻ allK (incr w)) (f k a))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: A ~k~> T B
mbinddt U G g ∘ mbind U f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbinddt (T k) G (g ◻ allK (incr w)) (f k a)) rewrite mbind_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: A ~k~> T B
mbinddt U G g
∘ mbinddt U (fun A : Type => A) (f ◻ allK extract) =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbinddt (T k) G (g ◻ allK (incr w)) (f k a))
    change (mbinddt U G g) with (map (F := fun A => A) (mbinddt U G g)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: A ~k~> T B
map (mbinddt U G g)
∘ mbinddt U (fun A : Type => A) (f ◻ allK extract) =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbinddt (T k) G (g ◻ allK (incr w)) (f k a))
    rewrite (dtp_mbinddt_mbinddt W T U (fun A => A) G).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: A ~k~> T B
mbinddt U ((fun A : Type => A) ∘ G)
  (g ⋆dtm (f ◻ allK extract)) =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbinddt (T k) G (g ◻ allK (incr w)) (f k a))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: A ~k~> T B
Mult_compose (fun A : Type => A) G = Mult_G0 now rewrite (Mult_compose_identity2 G).
  Qed.

  Lemma mbinddt_mmapd: forall
      (g: forall k, W * B -> G (T k C))
      (f: forall k, W * A -> B),
      mbinddt U G g ∘ mmapd U f =
        mbinddt U G (fun k '(w, a) => g k (w, f k (w, a))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : K -> W * A -> B),
mbinddt U G g ∘ mmapd U f =
mbinddt U G
  (fun (k : K) '(w, a) => g k (w, f k (w, a)))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : K -> W * A -> B),
mbinddt U G g ∘ mmapd U f =
mbinddt U G
  (fun (k : K) '(w, a) => g k (w, f k (w, a)))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> B
mbinddt U G g ∘ mmapd U f =
mbinddt U G
  (fun (k : K) '(w, a) => g k (w, f k (w, a)))
    erewrite mmapd_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> B
mbinddt U G g
∘ mbinddt U (fun A : Type => A) (mret T ◻ f) =
mbinddt U G
  (fun (k : K) '(w, a) => g k (w, f k (w, a)))
    change (mbinddt U G g) with (map (F := fun A => A) (mbinddt U G g)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> B
map (mbinddt U G g)
∘ mbinddt U (fun A : Type => A) (mret T ◻ f) =
mbinddt U G
  (fun (k : K) '(w, a) => g k (w, f k (w, a)))
    rewrite (dtp_mbinddt_mbinddt W T U (fun A => A) G).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> B
mbinddt U ((fun A : Type => A) ∘ G)
  (g ⋆dtm (mret T ◻ f)) =
mbinddt U G
  (fun (k : K) '(w, a) => g k (w, f k (w, a)))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> B
Mult_compose (fun A : Type => A) G = Mult_G0
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> B
g ⋆dtm (mret T ◻ f) =
(fun (k : K) '(w, a) => g k (w, f k (w, a))) now rewrite (Mult_compose_identity2 G).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> B
g ⋆dtm (mret T ◻ f) =
(fun (k : K) '(w, a) => g k (w, f k (w, a)))
    rewrite compose_dtm_lemma1.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> W * A -> B
(fun (k : K) '(w, a) => g k (w, f k (w, a))) =
(fun (k : K) '(w, a) => g k (w, f k (w, a)))
    reflexivity.
  Qed.

  Lemma mbinddt_mmapt: forall
      (g: forall k, W * B -> G (T k C))
      (f: K -> A -> F B),
      map (F := F) (mbinddt U G g) ∘ mmapt U F f =
        mbinddt U (F ∘ G)
          (fun k '(w, a) => map (F := F) (fun b => g k (w, b)) (f k a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : K -> A -> F B),
map (mbinddt U G g) ∘ mmapt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : K -> A -> F B),
map (mbinddt U G g) ∘ mmapt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> A -> F B
map (mbinddt U G g) ∘ mmapt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    rewrite mmapt_to_mmapdt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> A -> F B
map (mbinddt U G g) ∘ mmapdt U F (f ◻ allK extract) =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    rewrite mbinddt_mmapdt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> A -> F B
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) ((f ◻ allK extract) k (w, a))) =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    reflexivity.
  Qed.

  Lemma mbinddt_mmap: forall
      (g: forall k, W * B -> G (T k C))
      (f: K -> A -> B),
      mbinddt U G g ∘ mmap U f =
        mbinddt U G (fun k '(w, a) => g k (w, f k a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : K -> A -> B),
mbinddt U G g ∘ mmap U f =
mbinddt U G (fun (k : K) '(w, a) => g k (w, f k a))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
forall (g : forall k : K, W * B -> G (T k C))
  (f : K -> A -> B),
mbinddt U G g ∘ mmap U f =
mbinddt U G (fun (k : K) '(w, a) => g k (w, f k a))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> A -> B
mbinddt U G g ∘ mmap U f =
mbinddt U G (fun (k : K) '(w, a) => g k (w, f k a))
    rewrite mmap_to_mmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> A -> B
mbinddt U G g ∘ mmapd U (f ◻ allK extract) =
mbinddt U G (fun (k : K) '(w, a) => g k (w, f k a))
    rewrite mbinddt_mmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
A, B, C: Type
g: forall k : K, W * B -> G (T k C)
f: K -> A -> B
mbinddt U G
  (fun (k : K) '(w, a) =>
   g k (w, (f ◻ allK extract) k (w, a))) =
mbinddt U G (fun (k : K) '(w, a) => g k (w, f k a))
    reflexivity.
  Qed.

End derived_operations_composition.

(** ** Composition between Derived Operations *)
(** Composition laws involving one of <<mbindd>>/<<mmapdt>>/<<mbindt>>
    and another operation that is not a special cases. *)
(**********************************************************************)
Section mixed_composition_laws.

  Context
    (U: Type -> Type)
    (F G: Type -> Type)
    `{Applicative F}
    `{Applicative G}
    `{MultiDecoratedTraversablePreModule W T U}
    `{! MultiDecoratedTraversableMonad W T} {A B C: Type}.

  (** *** <<mbindd>> *)
  (** Operations with traversals are not special cases of <<mbindd>>. *)
  (********************************************************************)
  Lemma mbindd_mmapdt: forall
      (g: forall k, W * B -> T k C)
      (f: forall k, W * A -> F B),
      map (F := F) (mbindd U g) ∘ mmapdt U F f =
        mbinddt U F (fun (k: K) '(w, a) =>
                       map (F := F) (fun b => g k (w, b)) (f k (w, a))).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C) (f : K -> W * A -> F B),
map (mbindd U g) ∘ mmapdt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C) (f : K -> W * A -> F B),
map (mbindd U g) ∘ mmapdt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
    intros.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> F B
map (mbindd U g) ∘ mmapdt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a))) rewrite mmapdt_to_mbinddt.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> F B
map (mbindd U g) ∘ mbinddt U F (mapMret ◻ f) =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
    rewrite mbindd_to_mbinddt.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> F B
map (mbinddt U (fun A : Type => A) g)
∘ mbinddt U F (mapMret ◻ f) =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
    rewrite (dtp_mbinddt_mbinddt W T U F (fun x => x)).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> F B
mbinddt U (F ∘ (fun x : Type => x))
  (g ⋆dtm (mapMret ◻ f)) =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
    rewrite compose_dtm_lemma2.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> F B
mbinddt U (F ∘ (fun x : Type => x))
  (fun (k : K) '(w, a) =>
   map (g k ∘ pair w) (f k (w, a))) =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
    rewrite (Mult_compose_identity1 F).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> F B
mbinddt U (F ∘ (fun x : Type => x))
  (fun (k : K) '(w, a) =>
   map (g k ∘ pair w) (f k (w, a))) =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
    reflexivity.
  Qed.

  Lemma mbindd_mbindt: forall
      (g: forall k, W * B -> T k C)
      (f: forall k, A -> F (T k B)),
      map (F := F) (mbindd U g) ∘ mbindt U F f =
        mbinddt U F
          (fun (k: K) '(w, a) =>
             map (F := F) (mbindd (T k) (g ◻ allK (incr w))) (f k a)).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C)
  (f : forall k : K, A -> F (T k B)),
map (mbindd U g) ∘ mbindt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mbindd (T k) (g ◻ allK (incr w))) (f k a))
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C)
  (f : forall k : K, A -> F (T k B)),
map (mbindd U g) ∘ mbindt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mbindd (T k) (g ◻ allK (incr w))) (f k a))
    intros.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: forall k : K, A -> F (T k B)
map (mbindd U g) ∘ mbindt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mbindd (T k) (g ◻ allK (incr w))) (f k a))
    rewrite mbindd_to_mbinddt.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: forall k : K, A -> F (T k B)
map (mbinddt U (fun A : Type => A) g) ∘ mbindt U F f =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mbindd (T k) (g ◻ allK (incr w))) (f k a))
    rewrite mbindt_to_mbinddt.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: forall k : K, A -> F (T k B)
map (mbinddt U (fun A : Type => A) g)
∘ mbinddt U F (f ◻ allK extract) =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mbindd (T k) (g ◻ allK (incr w))) (f k a))
    rewrite (dtp_mbinddt_mbinddt W T U F (fun x => x)).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: forall k : K, A -> F (T k B)
mbinddt U (F ∘ (fun x : Type => x))
  (g ⋆dtm (f ◻ allK extract)) =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mbindd (T k) (g ◻ allK (incr w))) (f k a))
    rewrite (Mult_compose_identity1 F).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: forall k : K, A -> F (T k B)
mbinddt U (F ∘ (fun x : Type => x))
  (g ⋆dtm (f ◻ allK extract)) =
mbinddt U F
  (fun (k : K) '(w, a) =>
   map (mbindd (T k) (g ◻ allK (incr w))) (f k a))
    reflexivity.
  Qed.

  Lemma mbindd_mmapt: forall
      (g: forall k, W * B -> T k C)
      (f: forall k, A -> F B),
      map (F := F) (mbindd U g) ∘ mmapt U F f =
        mbinddt U F
          (fun (k: K) '(w, a) =>
             map (F := F) (fun b => g k (w, b)) (f k a)).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C) (f : K -> A -> F B),
map (mbindd U g) ∘ mmapt U F f =
mbinddt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C) (f : K -> A -> F B),
map (mbindd U g) ∘ mmapt U F f =
mbinddt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    intros.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> A -> F B
map (mbindd U g) ∘ mmapt U F f =
mbinddt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    rewrite mbindd_to_mbinddt.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> A -> F B
map (mbinddt U (fun A : Type => A) g) ∘ mmapt U F f =
mbinddt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    rewrite mmapt_to_mbinddt.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> A -> F B
map (mbinddt U (fun A : Type => A) g)
∘ mbinddt U F ((mapMret ◻ f) ◻ allK extract) =
mbinddt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    rewrite (dtp_mbinddt_mbinddt W T U F (fun x => x)).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> A -> F B
mbinddt U (F ∘ (fun x : Type => x))
  (g ⋆dtm ((mapMret ◻ f) ◻ allK extract)) =
mbinddt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    rewrite vec_compose_assoc.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> A -> F B
mbinddt U (F ∘ (fun x : Type => x))
  (g ⋆dtm (mapMret ◻ (f ◻ allK extract))) =
mbinddt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    rewrite compose_dtm_lemma2.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> A -> F B
mbinddt U (F ∘ (fun x : Type => x))
  (fun (k : K) '(w, a) =>
   map (g k ∘ pair w) ((f ◻ allK extract) k (w, a))) =
mbinddt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    rewrite (Mult_compose_identity1 F).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> A -> F B
mbinddt U (F ∘ (fun x : Type => x))
  (fun (k : K) '(w, a) =>
   map (g k ∘ pair w) ((f ◻ allK extract) k (w, a))) =
mbinddt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    reflexivity.
  Qed.

  (* TODO Also need to put <<mmapt_mbindd>> somewhere. *)

  (** *** <<mmapdt>> *)
  (** Operations with joins are not special cases of <<mmapdt>>. *)
  (********************************************************************)
  Lemma mmapdt_mbindd: forall
      (g: forall k, W * B -> G C)
      (f: forall k, W * A -> T k B),
      mmapdt U G g ∘ mbindd U f =
        mbinddt U G
          (fun (k: K) '(w, a) =>
             mmapdt (T k) G (g ◻ allK (incr w)) (f k (w, a))).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> G C) (f : W * A ~k~> T B),
mmapdt U G g ∘ mbindd U f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mmapdt (T k) G (g ◻ allK (incr w)) (f k (w, a)))
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> G C) (f : W * A ~k~> T B),
mmapdt U G g ∘ mbindd U f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mmapdt (T k) G (g ◻ allK (incr w)) (f k (w, a)))
    (* TODO *)
  Abort.

  Lemma mmapdt_mbindt: forall
      (g: forall k, W * B -> G C)
      (f: forall k, A -> F (T k B)),
      map (F := F) (mmapdt U G g) ∘ mbindt U F f =
        mbinddt U (F ∘ G)
          (fun (k: K) '(w, a) =>
             map (F := F) (mmapdt (T k) G (g ◻ allK (incr w))) (f k a)).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> G C)
  (f : forall k : K, A -> F (T k B)),
map (mmapdt U G g) ∘ mbindt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (mmapdt (T k) G (g ◻ allK (incr w))) (f k a))
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> G C)
  (f : forall k : K, A -> F (T k B)),
map (mmapdt U G g) ∘ mbindt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (mmapdt (T k) G (g ◻ allK (incr w))) (f k a))
    (* TODO *)
  Abort.

  Lemma mmapdt_mbind: forall
      (g: K -> W * B -> G C) (f: forall k, A -> T k B),
      mmapdt U G g ∘ mbind U f =
        mbinddt U G (fun k '(w, a) =>
                       mmapdt (T k) G (g ◻ allK (incr w)) (f k a)).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> G C) (f : A ~k~> T B),
mmapdt U G g ∘ mbind U f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mmapdt (T k) G (g ◻ allK (incr w)) (f k a))
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> G C) (f : A ~k~> T B),
mmapdt U G g ∘ mbind U f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mmapdt (T k) G (g ◻ allK (incr w)) (f k a))
    (* TODO *)
  Abort.

  (* TODO Also need to put <<mbind_mmapdt>> somewhere. *)

  (** *** <<mbindt>> *)
  (** Operations with decorations are not special cases of <<mbindt>>. *)
  (********************************************************************)
  Lemma mbindt_mbindd: forall
      (g: forall k, B -> G (T k C))
      (f: forall k, W * A -> T k B),
      mbindt U G g ∘ mbindd U f =
        mbinddt U G (fun (k: K) '(w, a) => mbindt (T k) G g (f k (w, a))).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : forall k : K, B -> G (T k C))
  (f : W * A ~k~> T B),
mbindt U G g ∘ mbindd U f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbindt (T k) G g (f k (w, a)))
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : forall k : K, B -> G (T k C))
  (f : W * A ~k~> T B),
mbindt U G g ∘ mbindd U f =
mbinddt U G
  (fun (k : K) '(w, a) =>
   mbindt (T k) G g (f k (w, a)))
    (* TODO *)
  Abort.

  Lemma mbindt_mmapdt: forall
      (g: forall k, B -> G (T k C))
      (f: forall k, W * A -> F B),
      map (F := F) (mbindt U G g) ∘ mmapdt U F f =
        mbinddt U (F ∘ G)
          (fun (k: K) '(w, a) => map (F := F) (g k) (f k (w, a))).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : forall k : K, B -> G (T k C))
  (f : K -> W * A -> F B),
map (mbindt U G g) ∘ mmapdt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) => map (g k) (f k (w, a)))
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : forall k : K, B -> G (T k C))
  (f : K -> W * A -> F B),
map (mbindt U G g) ∘ mmapdt U F f =
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) => map (g k) (f k (w, a)))
    (* TODO *)
  Abort.

  (* TODO <<mbindt_mmapd>> *)

  (* TODO Also need to put <<mmapd_mbindt>> somewhere. *)

End mixed_composition_laws.

(** ** Decorated Monad (<<mbindd>>) *)
(**********************************************************************)
Definition compose_dm
  `{ix: Index}
  {W: Type}
  {T: K -> Type -> Type}
  `{mn_op: Monoid_op W}
  `{mn_unit: Monoid_unit W}
  `{forall k, MBind W T (T k)}
  {A B C: Type}
  (g: forall k, W * B -> T k C)
  (f: forall k, W * A -> T k B): forall k, W * A -> T k C :=
  fun k '(w, a) => mbindd (T k) (g ◻ allK (incr w)) (f k (w, a)).

Infix "⋆dm" := compose_dm (at level 40).

Section DecoratedMonad.

  Context
    (U: Type -> Type)
    `{MultiDecoratedTraversablePreModule W T U}
    `{! MultiDecoratedTraversableMonad W T} {A B C: Type}.

  (** *** Composition and identity law *)
  (********************************************************************)
  Theorem mbindd_id:
    mbindd U (mret T ◻ allK extract) = @id (U A).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
mbindd U (mret T ◻ allK extract) = id
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
mbindd U (mret T ◻ allK extract) = id
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
mbindd U (mret T ◻ allK extract) = id rewrite mbindd_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
mbinddt U (fun A : Type => A) (mret T ◻ allK extract) =
id
    now rewrite <- (dtp_mbinddt_mret W T U).
  Qed.

  Theorem mbindd_mbindd: forall
      (g: W * B ~k~> T C)
      (f: W * A ~k~> T B),
      mbindd U g ∘ mbindd U f =
        mbindd U (fun (k: K) '(w, a) =>
                    mbindd (T k) (g ◻ allK (incr w)) (f k (w, a))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C) (f : W * A ~k~> T B),
mbindd U g ∘ mbindd U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) (g ◻ allK (incr w)) (f k (w, a)))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C) (f : W * A ~k~> T B),
mbindd U g ∘ mbindd U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) (g ◻ allK (incr w)) (f k (w, a)))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: W * A ~k~> T B
mbindd U g ∘ mbindd U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) (g ◻ allK (incr w)) (f k (w, a)))
    rewrite 3mbindd_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: W * A ~k~> T B
mbinddt U (fun A : Type => A) g
∘ mbinddt U (fun A : Type => A) f =
mbinddt U (fun A : Type => A)
  (fun (k : K) '(w, a) =>
   mbindd (T k) (g ◻ allK (incr w)) (f k (w, a)))
    change_left (map (F := fun x => x)
                   (mbinddt U (fun x: Type => x) g) ∘
                   (mbinddt U (fun x: Type => x) f)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: W * A ~k~> T B
map (mbinddt U (fun x : Type => x) g)
∘ mbinddt U (fun x : Type => x) f =
mbinddt U (fun A : Type => A)
  (fun (k : K) '(w, a) =>
   mbindd (T k) (g ◻ allK (incr w)) (f k (w, a)))
    rewrite (dtp_mbinddt_mbinddt W T U (fun x => x) (fun x => x)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: W * A ~k~> T B
mbinddt U ((fun x : Type => x) ∘ (fun x : Type => x))
  (g ⋆dtm f) =
mbinddt U (fun A : Type => A)
  (fun (k : K) '(w, a) =>
   mbindd (T k) (g ◻ allK (incr w)) (f k (w, a)))
    rewrite Mult_compose_identity2.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: W * A ~k~> T B
mbinddt U ((fun x : Type => x) ∘ (fun x : Type => x))
  (g ⋆dtm f) =
mbinddt U (fun A : Type => A)
  (fun (k : K) '(w, a) =>
   mbindd (T k) (g ◻ allK (incr w)) (f k (w, a)))
    reflexivity.
  Qed.

  (** *** Right unit law for monads *)
  (********************************************************************)
  Theorem mbindd_comp_mret: forall
      (k: K) (f: forall k, W * A -> T k B),
      mbindd (T k) f ∘ mret T k = f k ∘ ret (T := (W ×)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (k : K) (f : W * A ~k~> T B),
mbindd (T k) f ∘ mret T k = f k ∘ ret
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (k : K) (f : W * A ~k~> T B),
mbindd (T k) f ∘ mret T k = f k ∘ ret
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
k: K
f: W * A ~k~> T B
mbindd (T k) f ∘ mret T k = f k ∘ ret rewrite mbindd_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
k: K
f: W * A ~k~> T B
mbinddt (T k) (fun A : Type => A) f ∘ mret T k =
f k ∘ ret
    now rewrite (dtm_mbinddt_comp_mret W T k (fun A => A)).
  Qed.

  (** *** Composition with special cases on the right *)
  (** Composition with operations <<mbind>>/<<mmapd>>/<<mmap>> *)
  (********************************************************************)
  Lemma mbindd_mbind: forall
      (g: forall k, W * B -> T k C)
      (f: A ~k~> T B),
      mbindd U g ∘ mbind U f =
        mbindd U (fun (k: K) '(w, a) =>
                    mbindd (T k) (g ◻ allK (incr w)) (f k a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C) (f : A ~k~> T B),
mbindd U g ∘ mbind U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) (g ◻ allK (incr w)) (f k a))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C) (f : A ~k~> T B),
mbindd U g ∘ mbind U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) (g ◻ allK (incr w)) (f k a))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: A ~k~> T B
mbindd U g ∘ mbind U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) (g ◻ allK (incr w)) (f k a)) rewrite mbind_to_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: A ~k~> T B
mbindd U g ∘ mbindd U (f ◻ allK extract) =
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) (g ◻ allK (incr w)) (f k a))
    now rewrite mbindd_mbindd.
  Qed.

  Lemma mbindd_mmapd: forall
      (g: forall k, W * B -> T k C)
      (f: forall k, W * A -> B),
      mbindd U g ∘ mmapd U f =
        mbindd U (fun (k: K) '(w, a) => g k (w, f k (w, a))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C) (f : K -> W * A -> B),
mbindd U g ∘ mmapd U f =
mbindd U (fun (k : K) '(w, a) => g k (w, f k (w, a)))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C) (f : K -> W * A -> B),
mbindd U g ∘ mmapd U f =
mbindd U (fun (k : K) '(w, a) => g k (w, f k (w, a)))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> B
mbindd U g ∘ mmapd U f =
mbindd U (fun (k : K) '(w, a) => g k (w, f k (w, a))) rewrite mmapd_to_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> B
mbindd U g ∘ mbindd U (mret T ◻ f) =
mbindd U (fun (k : K) '(w, a) => g k (w, f k (w, a)))
    rewrite (mbindd_mbindd).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> B
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) (g ◻ allK (incr w))
     ((mret T ◻ f) k (w, a))) =
mbindd U (fun (k : K) '(w, a) => g k (w, f k (w, a)))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> B
(fun (k : K) '(w, a) =>
 mbindd (T k) (g ◻ allK (incr w))
   ((mret T ◻ f) k (w, a))) =
(fun (k : K) '(w, a) => g k (w, f k (w, a))) ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> B
k: K
w: W
a: A
mbindd (T k) (g ◻ allK (incr w))
  ((mret T ◻ f) k (w, a)) = g k (w, f k (w, a))
    unfold vec_compose, compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> B
k: K
w: W
a: A
mbindd (T k) (fun k : K => g k ○ allK (incr w) k)
  (mret T k (f k (w, a))) = g k (w, f k (w, a))
    compose near (f k (w, a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> B
k: K
w: W
a: A
(mbindd (T k) (fun k : K => g k ○ allK (incr w) k)
 ∘ mret T k) (f k (w, a)) =
(g k ∘ pair w) (f k (w, a))
    rewrite mbindd_to_mbinddt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> B
k: K
w: W
a: A
(mbinddt (T k) (fun A : Type => A)
   (fun k : K => g k ○ allK (incr w) k) ∘ mret T k)
  (f k (w, a)) = (g k ∘ pair w) (f k (w, a))
    rewrite (dtm_mbinddt_comp_mret W T k (fun A => A)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> B
k: K
w: W
a: A
(g k ○ allK (incr w) k ∘ pair Ƶ) (f k (w, a)) =
(g k ∘ pair w) (f k (w, a))
    unfold compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> W * A -> B
k: K
w: W
a: A
g k (w ● Ƶ, f k (w, a)) = g k (w, f k (w, a)) now simpl_monoid.
  Qed.

  Lemma mbindd_mmap: forall
      (g: forall k, W * B -> T k C)
      (f: forall k, A -> B),
      mbindd U g ∘ mmap U f =
        mbindd U (fun (k: K) '(w, a) => g k (w, f k a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C) (f : K -> A -> B),
mbindd U g ∘ mmap U f =
mbindd U (fun (k : K) '(w, a) => g k (w, f k a))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : W * B ~k~> T C) (f : K -> A -> B),
mbindd U g ∘ mmap U f =
mbindd U (fun (k : K) '(w, a) => g k (w, f k a))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> A -> B
mbindd U g ∘ mmap U f =
mbindd U (fun (k : K) '(w, a) => g k (w, f k a)) unfold mmap.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: W * B ~k~> T C
f: K -> A -> B
mbindd U g ∘ mmapd U (f ◻ allK extract) =
mbindd U (fun (k : K) '(w, a) => g k (w, f k a))
    now rewrite (mbindd_mmapd).
  Qed.

  (** *** Composition with special cases on the left *)
  (********************************************************************)
  Lemma mbind_mbindd: forall
      (g: forall k, B -> T k C)
      (f: forall k, W * A -> T k B),
      mbind U g ∘ mbindd U f =
        mbindd U (fun (k: K) => mbind (T k) g ∘ f k).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : B ~k~> T C) (f : W * A ~k~> T B),
mbind U g ∘ mbindd U f =
mbindd U (fun k : K => mbind (T k) g ∘ f k)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : B ~k~> T C) (f : W * A ~k~> T B),
mbind U g ∘ mbindd U f =
mbindd U (fun k : K => mbind (T k) g ∘ f k)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
mbind U g ∘ mbindd U f =
mbindd U (fun k : K => mbind (T k) g ∘ f k)
    rewrite mbind_to_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
mbindd U (g ◻ allK extract) ∘ mbindd U f =
mbindd U (fun k : K => mbind (T k) g ∘ f k)
    rewrite mbindd_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) ((g ◻ allK extract) ◻ allK (incr w))
     (f k (w, a))) =
mbindd U (fun k : K => mbind (T k) g ∘ f k)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
(fun (k : K) '(w, a) =>
 mbindd (T k) ((g ◻ allK extract) ◻ allK (incr w))
   (f k (w, a))) = (fun k : K => mbind (T k) g ∘ f k)
    ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
k: K
w: W
a: A
mbindd (T k) ((g ◻ allK extract) ◻ allK (incr w))
  (f k (w, a)) = (mbind (T k) g ∘ f k) (w, a)
    unfold mbind, compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
k: K
w: W
a: A
mbindd (T k) ((g ◻ allK extract) ◻ allK (incr w))
  (f k (w, a)) =
mbindd (T k) (g ◻ allK extract) (f k (w, a))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
k: K
w: W
a: A
(g ◻ allK extract) ◻ allK (incr w) = g ◻ allK extract
    now ext j [w2 b].
  Qed.

  Lemma mmapd_mbindd: forall
      (g: forall k, W * B -> C)
      (f: forall k, W * A -> T k B),
      mmapd U g ∘ mbindd U f =
        mbindd U (fun (k: K) '(w, a) =>
                    mmapd (T k) (g ◻ allK (incr w)) (f k (w, a))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> C) (f : W * A ~k~> T B),
mmapd U g ∘ mbindd U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mmapd (T k) (g ◻ allK (incr w)) (f k (w, a)))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> C) (f : W * A ~k~> T B),
mmapd U g ∘ mbindd U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mmapd (T k) (g ◻ allK (incr w)) (f k (w, a)))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: W * A ~k~> T B
mmapd U g ∘ mbindd U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mmapd (T k) (g ◻ allK (incr w)) (f k (w, a)))
    rewrite mmapd_to_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: W * A ~k~> T B
mbindd U (mret T ◻ g) ∘ mbindd U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mmapd (T k) (g ◻ allK (incr w)) (f k (w, a)))
    rewrite mbindd_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: W * A ~k~> T B
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) ((mret T ◻ g) ◻ allK (incr w))
     (f k (w, a))) =
mbindd U
  (fun (k : K) '(w, a) =>
   mmapd (T k) (g ◻ allK (incr w)) (f k (w, a)))
    reflexivity.
  Qed.

  Lemma mmap_mbindd: forall
      (g: forall k, B -> T k C)
      (f: forall k, W * A -> T k B),
      mbind U g ∘ mbindd U f =
        mbindd U (fun (k: K) => mbind (T k) g ∘ f k).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : B ~k~> T C) (f : W * A ~k~> T B),
mbind U g ∘ mbindd U f =
mbindd U (fun k : K => mbind (T k) g ∘ f k)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : B ~k~> T C) (f : W * A ~k~> T B),
mbind U g ∘ mbindd U f =
mbindd U (fun k : K => mbind (T k) g ∘ f k)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
mbind U g ∘ mbindd U f =
mbindd U (fun k : K => mbind (T k) g ∘ f k)
    rewrite mbind_to_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
mbindd U (g ◻ allK extract) ∘ mbindd U f =
mbindd U (fun k : K => mbind (T k) g ∘ f k)
    rewrite mbindd_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) ((g ◻ allK extract) ◻ allK (incr w))
     (f k (w, a))) =
mbindd U (fun k : K => mbind (T k) g ∘ f k)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
(fun (k : K) '(w, a) =>
 mbindd (T k) ((g ◻ allK extract) ◻ allK (incr w))
   (f k (w, a))) = (fun k : K => mbind (T k) g ∘ f k)
    ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
k: K
w: W
a: A
mbindd (T k) ((g ◻ allK extract) ◻ allK (incr w))
  (f k (w, a)) = (mbind (T k) g ∘ f k) (w, a) unfold compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
k: K
w: W
a: A
mbindd (T k) ((g ◻ allK extract) ◻ allK (incr w))
  (f k (w, a)) = mbind (T k) g (f k (w, a))
    rewrite mbind_to_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
k: K
w: W
a: A
mbindd (T k) ((g ◻ allK extract) ◻ allK (incr w))
  (f k (w, a)) =
mbindd (T k) (g ◻ allK extract) (f k (w, a)) fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: W * A ~k~> T B
k: K
w: W
a: A
(g ◻ allK extract) ◻ allK (incr w) = g ◻ allK extract
    now ext j [w2 b].
  Qed.

End DecoratedMonad.

(** ** Decorated Traversable Functor (<<mmapdt>>) *)
(**********************************************************************)
Section DecoratedTraversable.

  Context
    (U: Type -> Type)
    `{MultiDecoratedTraversablePreModule W T U}
    `{! MultiDecoratedTraversableMonad W T} {A B C: Type}
    `{Applicative F} `{Applicative G}.

  (** *** Composition and identity law *)
  (********************************************************************)
  Theorem mmapdt_id:
    mmapdt U (fun x => x) (allK extract) = @id (U A).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mmapdt U (fun x : Type => x) (allK extract) = id
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mmapdt U (fun x : Type => x) (allK extract) = id
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mmapdt U (fun x : Type => x) (allK extract) = id
    unfold mmapdt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mbinddt U (fun x : Type => x) (mapMret ◻ allK extract) =
id
    rewrite <- (dtp_mbinddt_mret W T U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mbinddt U (fun x : Type => x) (mapMret ◻ allK extract) =
mbinddt U (fun a : Type => a) (mret T ◻ allK extract)
    reflexivity.
  Qed.

  Theorem mmapdt_mmapdt: forall
      (g: forall k, W * B -> G C) (f: forall k, W * A -> F B),
      map (F := F) (mmapdt U G g) ∘ mmapdt U F f =
        mmapdt U (F ∘ G)
          (fun (k: K) '(w, a) =>
             map (F := F) (fun b => g k (w, b)) (f k (w, a))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (g : K -> W * B -> G C) (f : K -> W * A -> F B),
map (mmapdt U G g) ∘ mmapdt U F f =
mmapdt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (g : K -> W * B -> G C) (f : K -> W * A -> F B),
map (mmapdt U G g) ∘ mmapdt U F f =
mmapdt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
map (mmapdt U G g) ∘ mmapdt U F f =
mmapdt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
    unfold mmapdt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
map (mbinddt U G (mapMret ◻ g))
∘ mbinddt U F (mapMret ◻ f) =
mbinddt U (F ∘ G)
  (mapMret
   ◻ (fun (k : K) '(w, a) =>
      map (g k ○ pair w) (f k (w, a))))
    rewrite (dtp_mbinddt_mbinddt W T U F G).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
mbinddt U (F ∘ G) ((mapMret ◻ g) ⋆dtm (mapMret ◻ f)) =
mbinddt U (F ∘ G)
  (mapMret
   ◻ (fun (k : K) '(w, a) =>
      map (g k ○ pair w) (f k (w, a))))
    unfold compose_dtm.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
mbinddt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map
     (mbinddt (T k) G ((mapMret ◻ g) ◻ allK (incr w)))
     ((mapMret ◻ f) k (w, a))) =
mbinddt U (F ∘ G)
  (mapMret
   ◻ (fun (k : K) '(w, a) =>
      map (g k ○ pair w) (f k (w, a))))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
(fun (k : K) '(w, a) =>
 map (mbinddt (T k) G ((mapMret ◻ g) ◻ allK (incr w)))
   ((mapMret ◻ f) k (w, a))) =
mapMret
◻ (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a))) ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
k: K
w: W
a: A
map (mbinddt (T k) G ((mapMret ◻ g) ◻ allK (incr w)))
  ((mapMret ◻ f) k (w, a)) =
(mapMret
 ◻ (fun (k : K) '(w, a) =>
    map (g k ○ pair w) (f k (w, a)))) k (w, a)
    unfold vec_compose, mapMret, vec_apply, compose, allK, const.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
k: K
w: W
a: A
map
  (mbinddt (T k) G
     (fun (k : K) (a : W * B) =>
      map (mret T k) (g k (incr w a))))
  (map (mret T k) (f k (w, a))) =
map (mret T k) (map (g k ○ pair w) (f k (w, a)))
    compose near (f k (w, a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
k: K
w: W
a: A
(map
   (mbinddt (T k) G
      (fun (k : K) (a : W * B) =>
       map (mret T k) (g k (incr w a))))
 ∘ map (mret T k)) (f k (w, a)) =
(map (mret T k) ∘ map (g k ○ pair w)) (f k (w, a))
    rewrite (fun_map_map).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
k: K
w: W
a: A
map
  (mbinddt (T k) G
     (fun (k : K) (a : W * B) =>
      map (mret T k) (g k (incr w a))) ∘ mret T k)
  (f k (w, a)) =
(map (mret T k) ∘ map (g k ○ pair w)) (f k (w, a))
    unfold_ops @Map_compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
k: K
w: W
a: A
map
  (mbinddt (T k) G
     (fun (k : K) (a : W * B) =>
      map (mret T k) (g k (incr w a))) ∘ mret T k)
  (f k (w, a)) =
(map (map (mret T k)) ∘ map (g k ○ pair w))
  (f k (w, a))
    rewrite (fun_map_map).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
k: K
w: W
a: A
map
  (mbinddt (T k) G
     (fun (k : K) (a : W * B) =>
      map (mret T k) (g k (incr w a))) ∘ mret T k)
  (f k (w, a)) =
map (map (mret T k) ∘ (g k ○ pair w)) (f k (w, a))
    rewrite (dtm_mbinddt_comp_mret W T k G).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
k: K
w: W
a: A
map
  ((fun a : W * B => map (mret T k) (g k (incr w a)))
   ∘ pair Ƶ) (f k (w, a)) =
map (map (mret T k) ∘ (g k ○ pair w)) (f k (w, a))
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
k: K
w: W
a: A
(fun a : W * B => map (mret T k) (g k (incr w a)))
∘ pair Ƶ = map (mret T k) ∘ (g k ○ pair w) ext b.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
k: K
w: W
a: A
b: B
((fun a : W * B => map (mret T k) (g k (incr w a)))
 ∘ pair Ƶ) b = (map (mret T k) ∘ (g k ○ pair w)) b unfold compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
k: K
w: W
a: A
b: B
map (mret T k) (g k (incr w (Ƶ, b))) =
map (mret T k) (g k (w, b))
    compose near b.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
k: K
w: W
a: A
b: B
map (mret T k) (g k ((incr w ∘ pair Ƶ) b)) =
map (mret T k) ((g k ∘ pair w) b)
    erewrite pair_incr_zero.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: K -> W * B -> G C
f: K -> W * A -> F B
k: K
w: W
a: A
b: B
map (mret T k) (g k (w, b)) =
map (mret T k) ((g k ∘ pair w) b)
    now simpl_monoid.
  Qed.

  (** *** Composition with <<mret>> *)
  (********************************************************************)
  Lemma mmapdt_comp_mret: forall
      (k: K) (f: forall k, W * A -> F B),
      mmapdt (T k) F f ∘ mret T k = map (F := F) (mret T k) ∘ f k ∘ pair Ƶ.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (k : K) (f : K -> W * A -> F B),
mmapdt (T k) F f ∘ mret T k =
map (mret T k) ∘ f k ∘ pair Ƶ
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (k : K) (f : K -> W * A -> F B),
mmapdt (T k) F f ∘ mret T k =
map (mret T k) ∘ f k ∘ pair Ƶ
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
k: K
f: K -> W * A -> F B
mmapdt (T k) F f ∘ mret T k =
map (mret T k) ∘ f k ∘ pair Ƶ unfold mmapdt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
k: K
f: K -> W * A -> F B
mbinddt (T k) F (mapMret ◻ f) ∘ mret T k =
map (mret T k) ∘ f k ∘ pair Ƶ
    now rewrite (dtm_mbinddt_comp_mret W T k F).
  Qed.

  (** *** Purity *)
  (********************************************************************)
  Lemma mmapdt_pure:
    mmapdt U F (allK pure ◻ allK extract) = pure (A := U A).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mmapdt U F (allK pure ◻ allK extract) = pure
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mmapdt U F (allK pure ◻ allK extract) = pure
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mmapdt U F (allK pure ◻ allK extract) = pure
    unfold mmapdt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mbinddt U F (mapMret ◻ (allK pure ◻ allK extract)) =
pure
    rewrite <- vec_compose_assoc.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mbinddt U F ((mapMret ◻ allK pure) ◻ allK extract) =
pure
    replace (mapMret (A := A) ◻ allK pure) with
      ((fun k => pure (F := F)) ◻ mret (A := A) T).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mbinddt U F
  (((fun k : K => pure) ◻ mret T) ◻ allK extract) =
pure
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
(fun k : K => pure) ◻ mret T = mapMret ◻ allK pure
    {
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mbinddt U F
  (((fun k : K => pure) ◻ mret T) ◻ allK extract) =
pure rewrite vec_compose_assoc.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mbinddt U F
  ((fun k : K => pure) ◻ (mret T ◻ allK extract)) =
pure
      rewrite <- (dtp_mbinddt_morphism W T U (fun x => x) F (ϕ := @pure F _)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
pure
∘ mbinddt U (fun x : Type => x)
    (mret T ◻ allK extract) = pure
      now rewrite (dtp_mbinddt_mret W T U). }
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
(fun k : K => pure) ◻ mret T = mapMret ◻ allK pure
    {
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
(fun k : K => pure) ◻ mret T = mapMret ◻ allK pure unfold vec_compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
(fun k : K => pure ∘ mret T k) =
(fun k : K => mapMret k ∘ allK pure k) ext k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
k: K
pure ∘ mret T k = mapMret k ∘ allK pure k
      unfold mapMret, allK, const.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
k: K
pure ∘ mret T k =
vec_apply (fun k : K => map) (mret T) k ∘ pure
      ext a.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
k: K
a: A
(pure ∘ mret T k) a =
(vec_apply (fun k : K => map) (mret T) k ∘ pure) a unfold compose.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
k: K
a: A
pure (mret T k a) =
vec_apply (fun k : K => map) (mret T) k (pure a)
      rewrite <- app_pure_natural.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
k: K
a: A
map (mret T k) (pure a) =
vec_apply (fun k : K => map) (mret T) k (pure a)
      reflexivity. }
  Qed.

  (** *** Composition with special cases  on the right *)
  (********************************************************************)
  Lemma mmapdt_mmapt: forall
      (g: K -> W * B -> G C) (f: K -> A -> F B),
      map (F := F) (mmapdt U G g) ∘ mmapt U F f =
        mmapdt U (F ∘ G) (fun (k: K) '(w, a) => map (F := F) (fun b => g k (w, b)) (f k a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (g : K -> W * B -> G C) (f : K -> A -> F B),
map (mmapdt U G g) ∘ mmapt U F f =
mmapdt U (F ∘ G)
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (g : K -> W * B -> G C) (f : K -> A -> F B),
map (mmapdt U G g) ∘ mmapt U F f =
mmapdt U (F ∘ G)
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    (* TODO *)
  Abort.

  Lemma mmapdt_mmapd: forall
      (g: K -> W * B -> G C) (f: K -> W * A -> B),
      mmapdt U G g ∘ mmapd U f = mmapdt U G (fun k '(w, a) => g k (w, f k (w, a))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (g : K -> W * B -> G C) (f : K -> W * A -> B),
mmapdt U G g ∘ mmapd U f =
mmapdt U G
  (fun (k : K) '(w, a) => g k (w, f k (w, a)))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (g : K -> W * B -> G C) (f : K -> W * A -> B),
mmapdt U G g ∘ mmapd U f =
mmapdt U G
  (fun (k : K) '(w, a) => g k (w, f k (w, a)))
    (* TODO *)
  Abort.

  (* TODO <<mmapdt_mmap>> *)

  (** *** Composition with other operations on the left *)
  (********************************************************************)
  Lemma mmapd_mmapdt: forall
      (g: K -> W * B -> C) (f: K -> W * A -> F B),
      map (F := F) (mmapd U g) ∘ mmapdt U F f =
        mmapdt U F (fun k '(w, a) => map (F := F) (fun (b: B) => (g k (w, b))) (f k (w, a))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (g : K -> W * B -> C) (f : K -> W * A -> F B),
map (mmapd U g) ∘ mmapdt U F f =
mmapdt U F
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (g : K -> W * B -> C) (f : K -> W * A -> F B),
map (mmapd U g) ∘ mmapdt U F f =
mmapdt U F
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a)))
    (* TODO *)
  Abort.

  Lemma mmapt_mmapdt: forall
      (g: K -> B -> C) (f: K -> W * A -> F B),
      map (F := F) (mmap U g) ∘ mmapdt U F f = mmapdt U F (fun k => map (F := F) (g k) ∘ f k).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (g : K -> B -> C) (f : K -> W * A -> F B),
map (mmap U g) ∘ mmapdt U F f =
mmapdt U F (fun k : K => map (g k) ∘ f k)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
G: Type -> Type
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (g : K -> B -> C) (f : K -> W * A -> F B),
map (mmap U g) ∘ mmapdt U F f =
mmapdt U F (fun k : K => map (g k) ∘ f k)
    (* TODO *)
  Abort.

  (* TODO <<mmap_mmapdt>> *)

End DecoratedTraversable.

(** ** Traversable Monad (<<mbindt>>) *)
(**********************************************************************)
Section TraversableMonad.

  Context
    (U: Type -> Type)
    `{MultiDecoratedTraversablePreModule W T U}
    `{! MultiDecoratedTraversableMonad W T}
    {A B C: Type}
    (F G: Type -> Type)
    `{Applicative F} `{Applicative G}.

  (** *** Composition and identity law *)
  (********************************************************************)
  Theorem mbindt_id:
    mbindt U (fun x => x) (mret T) = @id (U A).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mbindt U (fun x : Type => x) (mret T) = id
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mbindt U (fun x : Type => x) (mret T) = id
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mbindt U (fun x : Type => x) (mret T) = id unfold mbindt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
mbinddt U (fun x : Type => x) (mret T ◻ allK extract) =
id
    now rewrite (dtp_mbinddt_mret W T U).
  Qed.

  Theorem mbindt_mbindt:
    forall (g: forall k, B -> G (T k C))
      (f: forall k, A -> F (T k B)),
      map (F := F) (mbindt U G g) ∘ mbindt U F f =
        mbindt U (F ∘ G) (fun (k: K) (a: A) => map (F := F) (mbindt (T k) G g) (f k a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (g : forall k : K, B -> G (T k C))
  (f : forall k : K, A -> F (T k B)),
map (mbindt U G g) ∘ mbindt U F f =
mbindt U (F ∘ G)
  (fun k : K => map (mbindt (T k) G g) ○ f k)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (g : forall k : K, B -> G (T k C))
  (f : forall k : K, A -> F (T k B)),
map (mbindt U G g) ∘ mbindt U F f =
mbindt U (F ∘ G)
  (fun k : K => map (mbindt (T k) G g) ○ f k)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: forall k : K, B -> G (T k C)
f: forall k : K, A -> F (T k B)
map (mbindt U G g) ∘ mbindt U F f =
mbindt U (F ∘ G)
  (fun k : K => map (mbindt (T k) G g) ○ f k) unfold mbindt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: forall k : K, B -> G (T k C)
f: forall k : K, A -> F (T k B)
map (mbinddt U G (g ◻ allK extract))
∘ mbinddt U F (f ◻ allK extract) =
mbinddt U (F ∘ G)
  ((fun k : K =>
    map (mbinddt (T k) G (g ◻ allK extract)) ○ f k)
   ◻ allK extract) rewrite (dtp_mbinddt_mbinddt W T U F G).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: forall k : K, B -> G (T k C)
f: forall k : K, A -> F (T k B)
mbinddt U (F ∘ G)
  ((g ◻ allK extract) ⋆dtm (f ◻ allK extract)) =
mbinddt U (F ∘ G)
  ((fun k : K =>
    map (mbinddt (T k) G (g ◻ allK extract)) ○ f k)
   ◻ allK extract)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: forall k : K, B -> G (T k C)
f: forall k : K, A -> F (T k B)
(g ◻ allK extract) ⋆dtm (f ◻ allK extract) =
(fun k : K =>
 map (mbinddt (T k) G (g ◻ allK extract)) ○ f k)
◻ allK extract ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: forall k : K, B -> G (T k C)
f: forall k : K, A -> F (T k B)
k: K
w: W
a: A
((g ◻ allK extract) ⋆dtm (f ◻ allK extract)) k (w, a) =
((fun k : K =>
  map (mbinddt (T k) G (g ◻ allK extract)) ○ f k)
 ◻ allK extract) k (w, a)
    unfold vec_compose, compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: forall k : K, B -> G (T k C)
f: forall k : K, A -> F (T k B)
k: K
w: W
a: A
map
  (mbinddt (T k) G
     ((fun k : K => g k ○ allK extract k)
      ◻ allK (incr w))) (f k a) =
map
  (mbinddt (T k) G (fun k : K => g k ○ allK extract k))
  (f k a)
    repeat fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: forall k : K, B -> G (T k C)
f: forall k : K, A -> F (T k B)
k: K
w: W
a: A
(fun k : K => g k ○ allK extract k) ◻ allK (incr w) =
(fun k : K => g k ○ allK extract k) ext k2 [w2 b].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
g: forall k : K, B -> G (T k C)
f: forall k : K, A -> F (T k B)
k: K
w: W
a: A
k2: K
w2: W
b: B
((fun k : K => g k ○ allK extract k) ◻ allK (incr w))
  k2 (w2, b) = g k2 (allK extract k2 (w2, b)) easy.
  Qed.

  (** *** Composition with <<mret>> *)
  (********************************************************************)
  Lemma mbindt_comp_mret:
    forall (k: K) (f: forall k, A -> F (T k B)),
      mbindt (T k) F f ∘ mret T k = f k.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (k : K) (f : forall k0 : K, A -> F (T k0 B)),
mbindt (T k) F f ∘ mret T k = f k
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
forall (k : K) (f : forall k0 : K, A -> F (T k0 B)),
mbindt (T k) F f ∘ mret T k = f k
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
k: K
f: forall k : K, A -> F (T k B)
mbindt (T k) F f ∘ mret T k = f k unfold mbindt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H2: Applicative G
k: K
f: forall k : K, A -> F (T k B)
mbinddt (T k) F (f ◻ allK extract) ∘ mret T k = f k
    now rewrite (dtm_mbinddt_comp_mret W T k F).
  Qed.

  (** *** Purity *)
  (********************************************************************)
  (* TODO *)

  (** *** Composition with special cases on the right *)
  (********************************************************************)
  (* TODO *)

  (** *** Composition with special cases on the left *)
  (********************************************************************)
  (* TODO *)

End TraversableMonad.

(** ** Heterogeneous Composition Laws *)
(** Composition laws between one of <<mbind>>/<<mmapd>>/<<mmapt>>
    and another operation, neither of which is a special case of the other. *)
(**********************************************************************)
Section mixed_composition_laws2.

  Context
    (U: Type -> Type)
    (F G: Type -> Type)
    `{Applicative F}
    `{Applicative G}
    `{MultiDecoratedTraversablePreModule W T U}
    `{! MultiDecoratedTraversableMonad W T} {A B C: Type}.

  (** *** <<mbind>> on the left *)
  (********************************************************************)
  Lemma mbind_mmapd:
    forall (g: forall k, B -> T k C)
      (f: K -> W * A -> B),
      mbind U g ∘ mmapd U f =
        mbindd U (g ◻ f).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : B ~k~> T C) (f : K -> W * A -> B),
mbind U g ∘ mmapd U f = mbindd U (g ◻ f)
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : B ~k~> T C) (f : K -> W * A -> B),
mbind U g ∘ mmapd U f = mbindd U (g ◻ f)
    intros.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> W * A -> B
mbind U g ∘ mmapd U f = mbindd U (g ◻ f)
    rewrite mmapd_to_mbindd.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> W * A -> B
mbind U g ∘ mbindd U (mret T ◻ f) = mbindd U (g ◻ f)
    rewrite mbind_to_mbindd.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> W * A -> B
mbindd U (g ◻ allK extract) ∘ mbindd U (mret T ◻ f) =
mbindd U (g ◻ f)
    rewrite (mbindd_mbindd U).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> W * A -> B
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) ((g ◻ allK extract) ◻ allK (incr w))
     ((mret T ◻ f) k (w, a))) = mbindd U (g ◻ f)
    fequal.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> W * A -> B
(fun (k : K) '(w, a) =>
 mbindd (T k) ((g ◻ allK extract) ◻ allK (incr w))
   ((mret T ◻ f) k (w, a))) = g ◻ f
    ext k [w a].
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> W * A -> B
k: K
w: W
a: A
mbindd (T k) ((g ◻ allK extract) ◻ allK (incr w))
  ((mret T ◻ f) k (w, a)) = (g ◻ f) k (w, a)
    unfold vec_compose, compose; cbn.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> W * A -> B
k: K
w: W
a: A
mbindd (T k)
  (fun (k : K) (a : W * B) =>
   g k (allK extract k (allK (incr w) k a)))
  (mret T k (f k (w, a))) = g k (f k (w, a))
    compose near (f k (w, a)) on left.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> W * A -> B
k: K
w: W
a: A
(mbindd (T k)
   (fun (k : K) (a : W * B) =>
    g k (allK extract k (allK (incr w) k a)))
 ∘ mret T k) (f k (w, a)) = g k (f k (w, a))
    now rewrite (mbindd_comp_mret (T := T)).
  Qed.

  Lemma mbind_mmapt:
    forall (g: forall k, B -> T k C)
      (f: K -> A -> F B),
      map (F := F) (mbind U g) ∘ mmapt U F f =
        mbindt U F (fun k => map (F := F) (g k) ∘ f k).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : B ~k~> T C) (f : K -> A -> F B),
map (mbind U g) ∘ mmapt U F f =
mbindt U F (fun k : K => map (g k) ∘ f k)
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : B ~k~> T C) (f : K -> A -> F B),
map (mbind U g) ∘ mmapt U F f =
mbindt U F (fun k : K => map (g k) ∘ f k)
    intros.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
map (mbind U g) ∘ mmapt U F f =
mbindt U F (fun k : K => map (g k) ∘ f k)
    rewrite (mmapt_to_mbindt U F).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
map (mbind U g) ∘ mbindt U F (mapMret ◻ f) =
mbindt U F (fun k : K => map (g k) ∘ f k)
    rewrite mbind_to_mbindt.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
map (mbindt U (fun A : Type => A) g)
∘ mbindt U F (mapMret ◻ f) =
mbindt U F (fun k : K => map (g k) ∘ f k)
    rewrite (mbindt_mbindt U F (fun A => A)).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
mbindt U (F ∘ (fun A : Type => A))
  (fun k : K =>
   map (mbindt (T k) (fun A : Type => A) g)
   ○ (mapMret ◻ f) k) =
mbindt U F (fun k : K => map (g k) ∘ f k)
    fequal.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
Mult_compose F (fun A : Type => A) = Mult_G
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
(fun k : K =>
 map (mbindt (T k) (fun A : Type => A) g)
 ○ (mapMret ◻ f) k) = 
(fun k : K => map (g k) ∘ f k)
    -
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
Mult_compose F (fun A : Type => A) = Mult_G now rewrite (Mult_compose_identity1 F).
    -
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
(fun k : K =>
 map (mbindt (T k) (fun A : Type => A) g)
 ○ (mapMret ◻ f) k) = (fun k : K => map (g k) ∘ f k) ext k a.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
k: K
a: A
map (mbindt (T k) (fun A : Type => A) g)
  ((mapMret ◻ f) k a) = (map (g k) ∘ f k) a unfold vec_compose, mapMret, vec_apply, compose.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
k: K
a: A
map (mbindt (T k) (fun A : Type => A) g)
  (map (mret T k) (f k a)) = map (g k) (f k a)
      compose near (f k a) on left.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
k: K
a: A
(map (mbindt (T k) (fun A : Type => A) g)
 ∘ map (mret T k)) (f k a) = map (g k) (f k a)
      rewrite fun_map_map.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
k: K
a: A
map (mbindt (T k) (fun A : Type => A) g ∘ mret T k)
  (f k a) = map (g k) (f k a)
      fequal.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
k: K
a: A
mbindt (T k) (fun A : Type => A) g ∘ mret T k = g k
      change (mbindt (T k) (fun A0: Type => A0) g)
        with (mbind (T k) g).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
k: K
a: A
mbind (T k) g ∘ mret T k = g k
      rewrite mbind_to_mbindd.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
k: K
a: A
mbindd (T k) (g ◻ allK extract) ∘ mret T k = g k
      rewrite mbindd_comp_mret.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: K -> A -> F B
k: K
a: A
(g ◻ allK extract) k ∘ ret = g k
      reflexivity.
  Qed.

  (** *** <<mmapd>> on the left *)
  (********************************************************************)
  Lemma mmapd_mbind:
    forall (g: K -> W * B -> C)
      (f: forall k, A -> T k B),
      mmapd U g ∘ mbind U f =
        mbindd U (fun k '(w, a) =>
                    mmapd (T k) (g ◻ allK (incr w)) (f k a)).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> C) (f : A ~k~> T B),
mmapd U g ∘ mbind U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mmapd (T k) (g ◻ allK (incr w)) (f k a))
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> C) (f : A ~k~> T B),
mmapd U g ∘ mbind U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mmapd (T k) (g ◻ allK (incr w)) (f k a))
    intros.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: A ~k~> T B
mmapd U g ∘ mbind U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mmapd (T k) (g ◻ allK (incr w)) (f k a)) rewrite mmapd_to_mbindd.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: A ~k~> T B
mbindd U (mret T ◻ g) ∘ mbind U f =
mbindd U
  (fun (k : K) '(w, a) =>
   mmapd (T k) (g ◻ allK (incr w)) (f k a)) rewrite mbind_to_mbindd.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: A ~k~> T B
mbindd U (mret T ◻ g) ∘ mbindd U (f ◻ allK extract) =
mbindd U
  (fun (k : K) '(w, a) =>
   mmapd (T k) (g ◻ allK (incr w)) (f k a))
    now rewrite (mbindd_mbindd U).
  Qed.

  Lemma mmapd_mmapt:
    forall (g: K -> W * B -> C)
      (f: forall k, A -> F B),
      map (F := F) (mmapd U g) ∘ mmapt U F f =
        mmapdt U F (fun k '(w, a) =>
                      map (F := F) (fun b => g k (w, b)) (f k a)).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> C) (f : K -> A -> F B),
map (mmapd U g) ∘ mmapt U F f =
mmapdt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> C) (f : K -> A -> F B),
map (mmapd U g) ∘ mmapt U F f =
mmapdt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    intros.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: K -> A -> F B
map (mmapd U g) ∘ mmapt U F f =
mmapdt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a)) rewrite mmapd_to_mmapdt.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: K -> A -> F B
map (mmapdt U (fun A : Type => A) g) ∘ mmapt U F f =
mmapdt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a)) rewrite mmapt_to_mmapdt.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: K -> A -> F B
map (mmapdt U (fun A : Type => A) g)
∘ mmapdt U F (f ◻ allK extract) =
mmapdt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a))
    rewrite (mmapdt_mmapdt U (G := fun A => A)).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: K -> A -> F B
mmapdt U (F ∘ (fun A : Type => A))
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) ((f ◻ allK extract) k (w, a))) =
mmapdt U F
  (fun (k : K) '(w, a) => map (g k ○ pair w) (f k a)) fequal.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: K -> A -> F B
Mult_compose F (fun A : Type => A) = Mult_G
    now rewrite (Mult_compose_identity1 F).
  Qed.

  (** *** <<mmapt>> on the left *)
  (********************************************************************)
  Lemma mmapt_mbind:
    forall (g: K -> B -> G C)
      (f: forall k, A -> T k B),
      mmapt U G g ∘ mbind U f =
        mbindt U G (fun k => mmapt (T k) G g ∘ f k).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> B -> G C) (f : A ~k~> T B),
mmapt U G g ∘ mbind U f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> B -> G C) (f : A ~k~> T B),
mmapt U G g ∘ mbind U f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    intros.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
mmapt U G g ∘ mbind U f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    rewrite mmapt_to_mbindt.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
mbindt U G (mapMret ◻ g) ∘ mbind U f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    rewrite mbind_to_mbindt.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
mbindt U G (mapMret ◻ g)
∘ mbindt U (fun A : Type => A) f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    unfold vec_compose, mapMret, vec_apply.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
mbindt U G (fun k : K => map (mret T k) ∘ g k)
∘ mbindt U (fun A : Type => A) f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    change (mbindt U G (fun k: K => map (F := G) (mret T k) ∘ g k))
      with (map (F := fun A => A)
              (mbindt U G (fun k: K => map (F := G) (mret T k) ∘ g k))).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
map (mbindt U G (fun k : K => map (mret T k) ∘ g k))
∘ mbindt U (fun A : Type => A) f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    rewrite (mbindt_mbindt U (fun A => A) G).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
mbindt U ((fun A : Type => A) ∘ G)
  (fun k : K =>
   map
     (mbindt (T k) G
        (fun k0 : K => map (mret T k0) ∘ g k0)) ○ 
   f k) =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    fequal.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
Mult_compose (fun A : Type => A) G = Mult_G0 now rewrite (Mult_compose_identity2 G).
  Qed.

  Lemma mmapt_mmapd:
    forall (g: K -> B -> G C)
      (f: forall k, A -> T k B),
      mmapt U G g ∘ mbind U f =
        mbindt U G (fun k => mmapt (T k) G g ∘ f k).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> B -> G C) (f : A ~k~> T B),
mmapt U G g ∘ mbind U f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
  Proof.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> B -> G C) (f : A ~k~> T B),
mmapt U G g ∘ mbind U f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    intros.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
mmapt U G g ∘ mbind U f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    rewrite mmapt_to_mbindt.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
mbindt U G (mapMret ◻ g) ∘ mbind U f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    rewrite mbind_to_mbindt.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
mbindt U G (mapMret ◻ g)
∘ mbindt U (fun A : Type => A) f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    unfold vec_compose, mapMret, vec_apply.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
mbindt U G (fun k : K => map (mret T k) ∘ g k)
∘ mbindt U (fun A : Type => A) f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    change (mbindt U G (fun k: K => map (F := G) (mret T k) ∘ g k))
      with (map (F := fun A => A)
              (mbindt U G (fun k: K => map (F := G) (mret T k) ∘ g k))).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
map (mbindt U G (fun k : K => map (mret T k) ∘ g k))
∘ mbindt U (fun A : Type => A) f =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    rewrite (mbindt_mbindt U (fun A => A) G).
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
mbindt U ((fun A : Type => A) ∘ G)
  (fun k : K =>
   map
     (mbindt (T k) G
        (fun k0 : K => map (mret T k0) ∘ g k0)) ○ 
   f k) =
mbindt U G (fun k : K => mmapt (T k) G g ∘ f k)
    fequal.
U, F, G: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H: Applicative F
Map_G0: Map G
Pure_G0: Pure G
Mult_G0: Mult G
H0: Applicative G
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H1: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H2: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> G C
f: A ~k~> T B
Mult_compose (fun A : Type => A) G = Mult_G0 now rewrite (Mult_compose_identity2 G).
  Qed.

End mixed_composition_laws2.

(** ** Monad (<<mbind>>) *)
(**********************************************************************)
Section Monad.

  Context
    (U: Type -> Type)
    `{MultiDecoratedTraversablePreModule W T U}
    `{! MultiDecoratedTraversableMonad W T}.

  (** *** Composition and identity law *)
  (********************************************************************)
  Theorem mbind_id: forall A,
      mbind U (fun k => mret T k) = @id (U A).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall A : Type, mbind U (fun k : K => mret T k) = id
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall A : Type, mbind U (fun k : K => mret T k) = id
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
mbind U (fun k : K => mret T k) = id rewrite mbind_to_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
mbindd U ((fun k : K => mret T k) ◻ allK extract) = id
    now rewrite <- (mbindd_id U).
  Qed.

  Theorem mbind_mbind {A B C}:
    forall (g: forall (k: K), B -> T k C) (f: forall (k: K), A -> T k B),
      mbind U g ∘ mbind U f =
        mbind U (fun (k: K) (a: A) => mbind (T k) g (f k a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : B ~k~> T C) (f : A ~k~> T B),
mbind U g ∘ mbind U f =
mbind U (fun k : K => mbind (T k) g ○ f k)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : B ~k~> T C) (f : A ~k~> T B),
mbind U g ∘ mbind U f =
mbind U (fun k : K => mbind (T k) g ○ f k)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: A ~k~> T B
mbind U g ∘ mbind U f =
mbind U (fun k : K => mbind (T k) g ○ f k) do 3 rewrite (mbind_to_mbindd U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: A ~k~> T B
mbindd U (g ◻ allK extract)
∘ mbindd U (f ◻ allK extract) =
mbindd U
  ((fun k : K => mbind (T k) g ○ f k) ◻ allK extract)
    rewrite (mbindd_mbindd U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: A ~k~> T B
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k) ((g ◻ allK extract) ◻ allK (incr w))
     ((f ◻ allK extract) k (w, a))) =
mbindd U
  ((fun k : K => mbind (T k) g ○ f k) ◻ allK extract)
    unfold vec_compose, compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: A ~k~> T B
mbindd U
  (fun (k : K) '(w, a) =>
   mbindd (T k)
     (fun (k0 : K) (a0 : W * B) =>
      g k0 (allK extract k0 (allK (incr w) k0 a0)))
     (f k a)) =
mbindd U
  (fun (k : K) (a : W * A) =>
   mbind (T k) g (f k (allK extract k a))) fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: A ~k~> T B
(fun (k : K) '(w, a) =>
 mbindd (T k)
   (fun (k0 : K) (a0 : W * B) =>
    g k0 (allK extract k0 (allK (incr w) k0 a0)))
   (f k a)) =
(fun (k : K) (a : W * A) =>
 mbind (T k) g (f k (allK extract k a)))
    ext k [w a].
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: A ~k~> T B
k: K
w: W
a: A
mbindd (T k)
  (fun (k : K) (a : W * B) =>
   g k (allK extract k (allK (incr w) k a))) (f k a) =
mbind (T k) g (f k (allK extract k (w, a)))
    rewrite (mbind_to_mbindd (T k)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: A ~k~> T B
k: K
w: W
a: A
mbindd (T k)
  (fun (k : K) (a : W * B) =>
   g k (allK extract k (allK (incr w) k a))) (f k a) =
mbindd (T k) (g ◻ allK extract)
  (f k (allK extract k (w, a)))
    cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: A ~k~> T B
k: K
w: W
a: A
mbindd (T k)
  (fun (k : K) (a : W * B) =>
   g k (allK extract k (allK (incr w) k a))) (f k a) =
mbindd (T k) (g ◻ allK extract) (f k a) fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: B ~k~> T C
f: A ~k~> T B
k: K
w: W
a: A
(fun (k : K) (a : W * B) =>
 g k (allK extract k (allK (incr w) k a))) =
g ◻ allK extract now ext j [w2 b].
  Qed.

  (** *** Composition with <<mret>> *)
  (********************************************************************)
  Lemma mbind_comp_mret {A B}:
    forall (k: K) (f: forall (k: K), A -> T k B) (a: A),
      mbind (T k) f (mret T k a) = f k a.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
forall (k : K) (f : A ~k~> T B) (a : A),
mbind (T k) f (mret T k a) = f k a
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
forall (k : K) (f : A ~k~> T B) (a : A),
mbind (T k) f (mret T k a) = f k a
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
k: K
f: A ~k~> T B
a: A
mbind (T k) f (mret T k a) = f k a rewrite mbind_to_mbindd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
k: K
f: A ~k~> T B
a: A
mbindd (T k) (f ◻ allK extract) (mret T k a) = f k a
    compose near a on left.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
k: K
f: A ~k~> T B
a: A
(mbindd (T k) (f ◻ allK extract) ∘ mret T k) a = f k a now rewrite mbindd_comp_mret.
  Qed.

  (* TODO <<mbind_mmap>> *)

  (* TODO <<mmap_mbind>> *)

End Monad.

(** ** Decorated Functor (<<mmapd>>) *)
(**********************************************************************)
Section DecoratedFunctor.

  Context
    (U: Type -> Type)
    `{MultiDecoratedTraversablePreModule W T U}
    `{! MultiDecoratedTraversableMonad W T}.

  (** *** Composition and identity law *)
  (********************************************************************)
  Theorem mmapd_id: forall A,
      mmapd U (allK extract) = @id (U A).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall A : Type, mmapd U (allK extract) = id
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall A : Type, mmapd U (allK extract) = id
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
mmapd U (allK extract) = id
    rewrite mmapd_to_mmapdt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
mmapdt U (fun A : Type => A) (allK extract) = id
    rewrite (mmapdt_id U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
id = id
    reflexivity.
  Qed.

  Theorem mmapd_mmapd {A B C}:
    forall (g: K -> W * B -> C) (f: K -> W * A -> B),
      mmapd U g ∘ mmapd U f =
        mmapd U (fun (k: K) '(w, a) => g k (w, f k (w, a))).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> C) (f : K -> W * A -> B),
mmapd U g ∘ mmapd U f =
mmapd U (fun (k : K) '(w, a) => g k (w, f k (w, a)))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> W * B -> C) (f : K -> W * A -> B),
mmapd U g ∘ mmapd U f =
mmapd U (fun (k : K) '(w, a) => g k (w, f k (w, a)))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: K -> W * A -> B
mmapd U g ∘ mmapd U f =
mmapd U (fun (k : K) '(w, a) => g k (w, f k (w, a))) do 3 rewrite mmapd_to_mmapdt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: K -> W * A -> B
mmapdt U (fun A : Type => A) g
∘ mmapdt U (fun A : Type => A) f =
mmapdt U (fun A : Type => A)
  (fun (k : K) '(w, a) => g k (w, f k (w, a)))
    change (mmapdt U (fun A => A) g) with
      (map (F := fun A => A) (mmapdt U (fun A => A) g)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: K -> W * A -> B
map (mmapdt U (fun A : Type => A) g)
∘ mmapdt U (fun A : Type => A) f =
mmapdt U (fun A : Type => A)
  (fun (k : K) '(w, a) => g k (w, f k (w, a)))
    rewrite (mmapdt_mmapdt U (G := fun A => A) (F := fun A => A)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: K -> W * A -> B
mmapdt U ((fun A : Type => A) ∘ (fun A : Type => A))
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a))) =
mmapdt U (fun A : Type => A)
  (fun (k : K) '(w, a) => g k (w, f k (w, a)))
    unfold compose; cbn.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: K -> W * A -> B
mmapdt U (fun a : Type => a)
  (fun (k : K) '(w, a) =>
   map (g k ○ pair w) (f k (w, a))) =
mmapdt U (fun A : Type => A)
  (fun (k : K) '(w, a) => g k (w, f k (w, a))) fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> W * B -> C
f: K -> W * A -> B
Mult_compose (fun A : Type => A) (fun A : Type => A) =
Mult_I
    now rewrite (Mult_compose_identity1 (fun A => A)).
  Qed.

  (** *** Composition with <<mret>> *)
  (********************************************************************)
  Lemma mmapd_comp_mret {A B}:
    forall (k: K) (f: K -> W * A -> B) (a: A),
      mmapd (T k) f (mret T k a) = mret T k (f k (Ƶ, a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
forall (k : K) (f : K -> W * A -> B) (a : A),
mmapd (T k) f (mret T k a) = mret T k (f k (Ƶ, a))
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
forall (k : K) (f : K -> W * A -> B) (a : A),
mmapd (T k) f (mret T k a) = mret T k (f k (Ƶ, a))
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
k: K
f: K -> W * A -> B
a: A
mmapd (T k) f (mret T k a) = mret T k (f k (Ƶ, a)) rewrite mmapd_to_mmapdt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
k: K
f: K -> W * A -> B
a: A
mmapdt (T k) (fun A : Type => A) f (mret T k a) =
mret T k (f k (Ƶ, a)) compose near a on left.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
k: K
f: K -> W * A -> B
a: A
(mmapdt (T k) (fun A : Type => A) f ∘ mret T k) a =
mret T k (f k (Ƶ, a))
    now rewrite (mmapdt_comp_mret (F := fun A => A)).
  Qed.

  (* TODO <<mmapd_mmap>> *)

  (* TODO <<mmap_mmapd>> *)

End DecoratedFunctor.

(** ** Traversable Functor (<<mmapt>>) *)
(**********************************************************************)
Section TraversableFunctor.

  Context
    (U: Type -> Type)
    `{MultiDecoratedTraversablePreModule W T U}
    `{! MultiDecoratedTraversableMonad W T}.

  (** *** Composition and identity law *)
  (********************************************************************)
  Theorem mmapt_id: forall A,
      mmapt U (fun A => A) (allK (@id A)) = @id (U A).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall A : Type,
mmapt U (fun A0 : Type => A0) (allK id) = id
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall A : Type,
mmapt U (fun A0 : Type => A0) (allK id) = id
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
mmapt U (fun A : Type => A) (allK id) = id unfold mmapt.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
mmapdt U (fun A : Type => A) (allK id ◻ allK extract) =
id
    now rewrite <- (mbindt_id U).
  Qed.

  Theorem mmapt_mmapt {A B C}:
    forall `{Applicative G} `{Applicative F}
      (g: K -> B -> G C) (f: K -> A -> F B),
      map (F := F) (mmapt U G g) ∘ mmapt U F f =
        mmapt U (F ∘ G) (fun (k: K) (a: A) => map (F := F) (g k) (f k a)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (F : Type -> Type) (Map_G0 : Map F)
  (Pure_G0 : Pure F) (Mult_G0 : Mult F),
Applicative F ->
forall (g : K -> B -> G C) (f : K -> A -> F B),
map (mmapt U G g) ∘ mmapt U F f =
mmapt U (F ∘ G) (fun k : K => map (g k) ○ f k)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (F : Type -> Type) (Map_G0 : Map F)
  (Pure_G0 : Pure F) (Mult_G0 : Mult F),
Applicative F ->
forall (g : K -> B -> G C) (f : K -> A -> F B),
map (mmapt U G g) ∘ mmapt U F f =
mmapt U (F ∘ G) (fun k : K => map (g k) ○ f k)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H1: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H2: Applicative F
g: K -> B -> G C
f: K -> A -> F B
map (mmapt U G g) ∘ mmapt U F f =
mmapt U (F ∘ G) (fun k : K => map (g k) ○ f k) rewrite (mmapt_to_mmapdt U G).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H1: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H2: Applicative F
g: K -> B -> G C
f: K -> A -> F B
map (mmapdt U G (g ◻ allK extract)) ∘ mmapt U F f =
mmapt U (F ∘ G) (fun k : K => map (g k) ○ f k)
    rewrite (mmapt_to_mmapdt U F).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H1: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H2: Applicative F
g: K -> B -> G C
f: K -> A -> F B
map (mmapdt U G (g ◻ allK extract))
∘ mmapdt U F (f ◻ allK extract) =
mmapt U (F ∘ G) (fun k : K => map (g k) ○ f k)
    rewrite (mmapt_to_mmapdt U (F ∘ G)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H1: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H2: Applicative F
g: K -> B -> G C
f: K -> A -> F B
map (mmapdt U G (g ◻ allK extract))
∘ mmapdt U F (f ◻ allK extract) =
mmapdt U (F ∘ G)
  ((fun k : K => map (g k) ○ f k) ◻ allK extract)
    rewrite (mmapdt_mmapdt U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H1: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H2: Applicative F
g: K -> B -> G C
f: K -> A -> F B
mmapdt U (F ∘ G)
  (fun (k : K) '(w, a) =>
   map ((g ◻ allK extract) k ○ pair w)
     ((f ◻ allK extract) k (w, a))) =
mmapdt U (F ∘ G)
  ((fun k : K => map (g k) ○ f k) ◻ allK extract)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H1: Applicative G
F: Type -> Type
Map_G0: Map F
Pure_G0: Pure F
Mult_G0: Mult F
H2: Applicative F
g: K -> B -> G C
f: K -> A -> F B
(fun (k : K) '(w, a) =>
 map ((g ◻ allK extract) k ○ pair w)
   ((f ◻ allK extract) k (w, a))) =
(fun k : K => map (g k) ○ f k) ◻ allK extract now ext k [w a].
  Qed.

  (** *** Composition with <<mret>> *)
  (********************************************************************)
  Lemma mmapt_comp_mret {A B}:
    forall  `{Applicative F} (k: K) (f: K -> A -> F B) (a: A),
      mmapt (T k) F f (mret T k a) = map (F := F) (mret T k) (f k a).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
forall (F : Type -> Type) (Map_G : Map F)
  (Pure_G : Pure F) (Mult_G : Mult F),
Applicative F ->
forall (k : K) (f : K -> A -> F B) (a : A),
mmapt (T k) F f (mret T k a) = map (mret T k) (f k a)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
forall (F : Type -> Type) (Map_G : Map F)
  (Pure_G : Pure F) (Mult_G : Mult F),
Applicative F ->
forall (k : K) (f : K -> A -> F B) (a : A),
mmapt (T k) F f (mret T k a) = map (mret T k) (f k a)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
k: K
f: K -> A -> F B
a: A
mmapt (T k) F f (mret T k a) = map (mret T k) (f k a) rewrite (mmapt_to_mmapdt (T k)).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
k: K
f: K -> A -> F B
a: A
mmapdt (T k) F (f ◻ allK extract) (mret T k a) =
map (mret T k) (f k a) compose near a on left.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
F: Type -> Type
Map_G: Map F
Pure_G: Pure F
Mult_G: Mult F
H1: Applicative F
k: K
f: K -> A -> F B
a: A
(mmapdt (T k) F (f ◻ allK extract) ∘ mret T k) a =
map (mret T k) (f k a)
    now rewrite mmapdt_comp_mret.
  Qed.

  (* TODO <<mmapt_mmap>> *)

  (* TODO <<mmap_mmapt>> *)

End TraversableFunctor.

(** ** Functor (<<mmap>>) *)
(**********************************************************************)
Section Functor.

  Context
    (U: Type -> Type)
    `{MultiDecoratedTraversablePreModule W T U}
    `{! MultiDecoratedTraversableMonad W T}.

  (** *** Composition and identity law *)
  (********************************************************************)
  Theorem mmap_id: forall A,
      mmap U (allK (@id A)) = @id (U A).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall A : Type, mmap U (allK id) = id
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
forall A : Type, mmap U (allK id) = id
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A: Type
mmap U (allK id) = id apply (dtp_mbinddt_mret W T U).
  Qed.

  Theorem mmap_mmap {A B C}: forall
      (g: K -> B -> C) (f: K -> A -> B),
      mmap U g ∘ mmap U f = mmap U (g ◻ f).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> B -> C) (f : K -> A -> B),
mmap U g ∘ mmap U f = mmap U (g ◻ f)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
forall (g : K -> B -> C) (f : K -> A -> B),
mmap U g ∘ mmap U f = mmap U (g ◻ f)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> C
f: K -> A -> B
mmap U g ∘ mmap U f = mmap U (g ◻ f) do 3 rewrite mmap_to_mmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> C
f: K -> A -> B
mmapd U (g ◻ allK extract)
∘ mmapd U (f ◻ allK extract) =
mmapd U ((g ◻ f) ◻ allK extract)
    rewrite (mmapd_mmapd U).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> C
f: K -> A -> B
mmapd U
  (fun (k : K) '(w, a) =>
   (g ◻ allK extract) k
     (w, (f ◻ allK extract) k (w, a))) =
mmapd U ((g ◻ f) ◻ allK extract)
    fequal.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B, C: Type
g: K -> B -> C
f: K -> A -> B
(fun (k : K) '(w, a) =>
 (g ◻ allK extract) k (w, (f ◻ allK extract) k (w, a))) =
(g ◻ f) ◻ allK extract now ext k [w a].
  Qed.

  (** *** Composition with <<mret>> *)
  (********************************************************************)
  Lemma mmap_comp_mret {A B}:
    forall (f: K -> A -> B) (a: A) (k: K),
      mmap (T k) f (mret T k a) = mret T k (f k a).
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
forall (f : K -> A -> B) (a : A) (k : K),
mmap (T k) f (mret T k a) = mret T k (f k a)
  Proof.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
forall (f : K -> A -> B) (a : A) (k : K),
mmap (T k) f (mret T k a) = mret T k (f k a)
    intros.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> A -> B
a: A
k: K
mmap (T k) f (mret T k a) = mret T k (f k a) rewrite mmap_to_mmapd.
U: Type -> Type
ix: Index
W: Type
T: K -> Type -> Type
MReturn0: MReturn T
MBind0: MBind W T U
H: forall k : K, MBind W T (T k)
mn_op: Monoid_op W
mn_unit: Monoid_unit W
H0: MultiDecoratedTraversablePreModule W T U
MultiDecoratedTraversableMonad0: MultiDecoratedTraversableMonad
  W T
A, B: Type
f: K -> A -> B
a: A
k: K
mmapd (T k) (f ◻ allK extract) (mret T k a) =
mret T k (f k a)
    now rewrite mmapd_comp_mret.
  Qed.

End Functor.